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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem26 Structured version   Visualization version   Unicode version
Theorem poimirlem26 33435
Description: Lemma for poimir 33442 showing an even difference between the number of admissible faces and the number of admissible simplices. Equation (6) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 |- ( ph -> N e. NN )
poimirlem28.1 |- ( p = ( ( 1st ` s ) oF + ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = C )
poimirlem28.2 |- ( ( ph /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> B e. ( 0 ... N ) )
Assertion
Ref Expression
poimirlem26 |- ( ph -> 2 || ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) ) )
Distinct variable groups:   f, i, j, p, s, t   ph, j   j, N   ph, i, p, s, t   B, f, i, j, s, t   f, K, i, j, p, s, t   f, N, i, p, s, t   C, i, p, t
Allowed substitution hints:   ph( f)   B( p)   C( f, j, s)
Proof of Theorem poimirlem26
Dummy variables k x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzofi 12773 . . . . . 6 |- ( 0..^ K ) e. Fin
2 fzfi 12771 . . . . . 6 |- ( 1 ... N ) e. Fin
3 mapfi 8262 . . . . . 6 |- ( ( ( 0..^ K ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( 0..^ K ) ^m ( 1 ... N ) ) e. Fin )
41, 2, 3mp2an 708 . . . . 5 |- ( ( 0..^ K ) ^m ( 1 ... N ) ) e. Fin
5 mapfi 8262 . . . . . . 7 |- ( ( ( 1 ... N ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( 1 ... N ) ^m ( 1 ... N ) ) e. Fin )
62, 2, 5mp2an 708 . . . . . 6 |- ( ( 1 ... N ) ^m ( 1 ... N ) ) e. Fin
7 f1of 6137 . . . . . . . 8 |- ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> f : ( 1 ... N ) --> ( 1 ... N ) )
87ss2abi 3674 . . . . . . 7 |- { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } C_ { f | f : ( 1 ... N ) --> ( 1 ... N ) }
9 ovex 6678 . . . . . . . 8 |- ( 1 ... N ) e. _V
109, 9mapval 7869 . . . . . . 7 |- ( ( 1 ... N ) ^m ( 1 ... N ) ) = { f | f : ( 1 ... N ) --> ( 1 ... N ) }
118, 10sseqtr4i 3638 . . . . . 6 |- { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } C_ ( ( 1 ... N ) ^m ( 1 ... N ) )
12 ssfi 8180 . . . . . 6 |- ( ( ( ( 1 ... N ) ^m ( 1 ... N ) ) e. Fin /\ { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } C_ ( ( 1 ... N ) ^m ( 1 ... N ) ) ) -> { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } e. Fin )
136, 11, 12mp2an 708 . . . . 5 |- { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } e. Fin
144, 13pm3.2i 471 . . . 4 |- ( ( ( 0..^ K ) ^m ( 1 ... N ) ) e. Fin /\ { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } e. Fin )
15 xpfi 8231 . . . 4 |- ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) e. Fin /\ { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } e. Fin ) -> ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin )
1614, 15mp1i 13 . . 3 |- ( ph -> ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin )
17 2z 11409 . . . 4 |- 2 e. ZZ
1817a1i 11 . . 3 |- ( ph -> 2 e. ZZ )
19 snfi 8038 . . . . . . 7 |- { x } e. Fin
20 fzfi 12771 . . . . . . . 8 |- ( 0 ... N ) e. Fin
21 rabfi 8185 . . . . . . . 8 |- ( ( 0 ... N ) e. Fin -> { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } e. Fin )
2220, 21ax-mp 5 . . . . . . 7 |- { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } e. Fin
23 xpfi 8231 . . . . . . 7 |- ( ( { x } e. Fin /\ { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } e. Fin ) -> ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) e. Fin )
2419, 22, 23mp2an 708 . . . . . 6 |- ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) e. Fin
25 hashcl 13147 . . . . . 6 |- ( ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) e. Fin -> ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) e. NN0 )
2624, 25ax-mp 5 . . . . 5 |- ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) e. NN0
2726nn0zi 11402 . . . 4 |- ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) e. ZZ
2827a1i 11 . . 3 |- ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) -> ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) e. ZZ )
29 poimir.0 . . . . . . . . 9 |- ( ph -> N e. NN )
3029ad2antrr 762 . . . . . . . 8 |- ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> N e. NN )
31 nfv 1843 . . . . . . . . . 10 |- F/ j p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) ) )
32 nfcsb1v 3549 . . . . . . . . . . 11 |- F/_ j [_ k / j ]_ [_ t / s ]_ C
3332nfeq2 2780 . . . . . . . . . 10 |- F/ j B = [_ k / j ]_ [_ t / s ]_ C
3431, 33nfim 1825 . . . . . . . . 9 |- F/ j ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = [_ k / j ]_ [_ t / s ]_ C )
35 oveq2 6658 . . . . . . . . . . . . . . 15 |- ( j = k -> ( 1 ... j ) = ( 1 ... k ) )
3635imaeq2d 5466 . . . . . . . . . . . . . 14 |- ( j = k -> ( ( 2nd ` t ) " ( 1 ... j ) ) = ( ( 2nd ` t ) " ( 1 ... k ) ) )
3736xpeq1d 5138 . . . . . . . . . . . . 13 |- ( j = k -> ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) )
38 oveq1 6657 . . . . . . . . . . . . . . . 16 |- ( j = k -> ( j + 1 ) = ( k + 1 ) )
3938oveq1d 6665 . . . . . . . . . . . . . . 15 |- ( j = k -> ( ( j + 1 ) ... N ) = ( ( k + 1 ) ... N ) )
4039imaeq2d 5466 . . . . . . . . . . . . . 14 |- ( j = k -> ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) )
4140xpeq1d 5138 . . . . . . . . . . . . 13 |- ( j = k -> ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) )
4237, 41uneq12d 3768 . . . . . . . . . . . 12 |- ( j = k -> ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) ) )
4342oveq2d 6666 . . . . . . . . . . 11 |- ( j = k -> ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) ) ) )
4443eqeq2d 2632 . . . . . . . . . 10 |- ( j = k -> ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) <-> p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) ) ) ) )
45 csbeq1a 3542 . . . . . . . . . . 11 |- ( j = k -> [_ t / s ]_ C = [_ k / j ]_ [_ t / s ]_ C )
4645eqeq2d 2632 . . . . . . . . . 10 |- ( j = k -> ( B = [_ t / s ]_ C <-> B = [_ k / j ]_ [_ t / s ]_ C ) )
4744, 46imbi12d 334 . . . . . . . . 9 |- ( j = k -> ( ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = [_ t / s ]_ C ) <-> ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = [_ k / j ]_ [_ t / s ]_ C ) ) )
48 nfv 1843 . . . . . . . . . . 11 |- F/ s p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) )
49 nfcsb1v 3549 . . . . . . . . . . . 12 |- F/_ s [_ t / s ]_ C
5049nfeq2 2780 . . . . . . . . . . 11 |- F/ s B = [_ t / s ]_ C
5148, 50nfim 1825 . . . . . . . . . 10 |- F/ s ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = [_ t / s ]_ C )
52 fveq2 6191 . . . . . . . . . . . . 13 |- ( s = t -> ( 1st ` s ) = ( 1st ` t ) )
53 fveq2 6191 . . . . . . . . . . . . . . . 16 |- ( s = t -> ( 2nd ` s ) = ( 2nd ` t ) )
5453imaeq1d 5465 . . . . . . . . . . . . . . 15 |- ( s = t -> ( ( 2nd ` s ) " ( 1 ... j ) ) = ( ( 2nd ` t ) " ( 1 ... j ) ) )
5554xpeq1d 5138 . . . . . . . . . . . . . 14 |- ( s = t -> ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) )
5653imaeq1d 5465 . . . . . . . . . . . . . . 15 |- ( s = t -> ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) )
5756xpeq1d 5138 . . . . . . . . . . . . . 14 |- ( s = t -> ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
5855, 57uneq12d 3768 . . . . . . . . . . . . 13 |- ( s = t -> ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) )
5952, 58oveq12d 6668 . . . . . . . . . . . 12 |- ( s = t -> ( ( 1st ` s ) oF + ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
6059eqeq2d 2632 . . . . . . . . . . 11 |- ( s = t -> ( p = ( ( 1st ` s ) oF + ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) <-> p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) )
61 csbeq1a 3542 . . . . . . . . . . . 12 |- ( s = t -> C = [_ t / s ]_ C )
6261eqeq2d 2632 . . . . . . . . . . 11 |- ( s = t -> ( B = C <-> B = [_ t / s ]_ C ) )
6360, 62imbi12d 334 . . . . . . . . . 10 |- ( s = t -> ( ( p = ( ( 1st ` s ) oF + ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = C ) <-> ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = [_ t / s ]_ C ) ) )
64 poimirlem28.1 . . . . . . . . . 10 |- ( p = ( ( 1st ` s ) oF + ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = C )
6551, 63, 64chvar 2262 . . . . . . . . 9 |- ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = [_ t / s ]_ C )
6634, 47, 65chvar 2262 . . . . . . . 8 |- ( p = ( ( 1st ` t ) oF + ( ( ( ( 2nd ` t ) " ( 1 ... k ) ) X. { 1 } ) u. ( ( ( 2nd ` t ) " ( ( k + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = [_ k / j ]_ [_ t / s ]_ C )
67 poimirlem28.2 . . . . . . . . 9 |- ( ( ph /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> B e. ( 0 ... N ) )
6867ad4ant14 1293 . . . . . . . 8 |- ( ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> B e. ( 0 ... N ) )
69 xp1st 7198 . . . . . . . . . 10 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` x ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) )
70 elmapi 7879 . . . . . . . . . 10 |- ( ( 1st ` x ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` x ) : ( 1 ... N ) --> ( 0..^ K ) )
7169, 70syl 17 . . . . . . . . 9 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` x ) : ( 1 ... N ) --> ( 0..^ K ) )
7271ad2antlr 763 . . . . . . . 8 |- ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> ( 1st ` x ) : ( 1 ... N ) --> ( 0..^ K ) )
73 xp2nd 7199 . . . . . . . . . 10 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` x ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
74 fvex 6201 . . . . . . . . . . 11 |- ( 2nd ` x ) e. _V
75 f1oeq1 6127 . . . . . . . . . . 11 |- ( f = ( 2nd ` x ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` x ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) )
7674, 75elab 3350 . . . . . . . . . 10 |- ( ( 2nd ` x ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` x ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
7773, 76sylib 208 . . . . . . . . 9 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` x ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
7877ad2antlr 763 . . . . . . . 8 |- ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> ( 2nd ` x ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
79 nfcv 2764 . . . . . . . . . . . . 13 |- F/_ j N
80 nfcv 2764 . . . . . . . . . . . . . 14 |- F/_ j x
8180, 32nfcsb 3551 . . . . . . . . . . . . 13 |- F/_ j [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C
8279, 81nfne 2894 . . . . . . . . . . . 12 |- F/ j N =/= [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C
83 nfcv 2764 . . . . . . . . . . . . . . 15 |- F/_ t C
8483, 49, 61cbvcsb 3538 . . . . . . . . . . . . . 14 |- [_ x / s ]_ C = [_ x / t ]_ [_ t / s ]_ C
8545csbeq2dv 3992 . . . . . . . . . . . . . 14 |- ( j = k -> [_ x / t ]_ [_ t / s ]_ C = [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C )
8684, 85syl5eq 2668 . . . . . . . . . . . . 13 |- ( j = k -> [_ x / s ]_ C = [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C )
8786neeq2d 2854 . . . . . . . . . . . 12 |- ( j = k -> ( N =/= [_ x / s ]_ C <-> N =/= [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C ) )
8882, 87rspc 3303 . . . . . . . . . . 11 |- ( k e. ( 0 ... N ) -> ( A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> N =/= [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C ) )
8988impcom 446 . . . . . . . . . 10 |- ( ( A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C /\ k e. ( 0 ... N ) ) -> N =/= [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C )
9089adantll 750 . . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) /\ k e. ( 0 ... N ) ) -> N =/= [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C )
91 1st2nd2 7205 . . . . . . . . . . 11 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
9291csbeq1d 3540 . . . . . . . . . 10 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C )
9392ad3antlr 767 . . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) /\ k e. ( 0 ... N ) ) -> [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C )
9490, 93neeqtrd 2863 . . . . . . . 8 |- ( ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) /\ k e. ( 0 ... N ) ) -> N =/= [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C )
9530, 66, 68, 72, 78, 94poimirlem25 33434 . . . . . . 7 |- ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> 2 || ( # ` { y e. ( 0 ... N ) | A. i e. ( 0 ... ( N - 1 ) ) E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C } ) )
96 nfv 1843 . . . . . . . . . . . . . 14 |- F/ k i = [_ x / s ]_ C
9781nfeq2 2780 . . . . . . . . . . . . . 14 |- F/ j i = [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C
9886eqeq2d 2632 . . . . . . . . . . . . . 14 |- ( j = k -> ( i = [_ x / s ]_ C <-> i = [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C ) )
9996, 97, 98cbvrex 3168 . . . . . . . . . . . . 13 |- ( E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C <-> E. k e. ( ( 0 ... N ) \ { y } ) i = [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C )
10092eqeq2d 2632 . . . . . . . . . . . . . 14 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( i = [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C <-> i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C ) )
101100rexbidv 3052 . . . . . . . . . . . . 13 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( E. k e. ( ( 0 ... N ) \ { y } ) i = [_ x / t ]_ [_ k / j ]_ [_ t / s ]_ C <-> E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C ) )
10299, 101syl5rbb 273 . . . . . . . . . . . 12 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C <-> E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C ) )
103102ralbidv 2986 . . . . . . . . . . 11 |- ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( A. i e. ( 0 ... ( N - 1 ) ) E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C ) )
104 iba 524 . . . . . . . . . . 11 |- ( A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) )
105103, 104sylan9bb 736 . . . . . . . . . 10 |- ( ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> ( A. i e. ( 0 ... ( N - 1 ) ) E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) )
106105rabbidv 3189 . . . . . . . . 9 |- ( ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> { y e. ( 0 ... N ) | A. i e. ( 0 ... ( N - 1 ) ) E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C } = { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } )
107106fveq2d 6195 . . . . . . . 8 |- ( ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> ( # ` { y e. ( 0 ... N ) | A. i e. ( 0 ... ( N - 1 ) ) E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C } ) = ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
108107adantll 750 . . . . . . 7 |- ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> ( # ` { y e. ( 0 ... N ) | A. i e. ( 0 ... ( N - 1 ) ) E. k e. ( ( 0 ... N ) \ { y } ) i = [_ <. ( 1st ` x ) , ( 2nd ` x ) >. / t ]_ [_ k / j ]_ [_ t / s ]_ C } ) = ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
10995, 108breqtrd 4679 . . . . . 6 |- ( ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> 2 || ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
110109ex 450 . . . . 5 |- ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) -> ( A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> 2 || ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) )
111 dvds0 14997 . . . . . . . 8 |- ( 2 e. ZZ -> 2 || 0 )
11217, 111ax-mp 5 . . . . . . 7 |- 2 || 0
113 hash0 13158 . . . . . . 7 |- ( # ` (/) ) = 0
114112, 113breqtrri 4680 . . . . . 6 |- 2 || ( # ` (/) )
115 simpr 477 . . . . . . . . . 10 |- ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) -> A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C )
116115con3i 150 . . . . . . . . 9 |- ( -. A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> -. ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) )
117116ralrimivw 2967 . . . . . . . 8 |- ( -. A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> A. y e. ( 0 ... N ) -. ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) )
118 rabeq0 3957 . . . . . . . 8 |- ( { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } = (/) <-> A. y e. ( 0 ... N ) -. ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) )
119117, 118sylibr 224 . . . . . . 7 |- ( -. A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } = (/) )
120119fveq2d 6195 . . . . . 6 |- ( -. A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) = ( # ` (/) ) )
121114, 120syl5breqr 4691 . . . . 5 |- ( -. A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C -> 2 || ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
122110, 121pm2.61d1 171 . . . 4 |- ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) -> 2 || ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
123 hashxp 13221 . . . . . 6 |- ( ( { x } e. Fin /\ { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } e. Fin ) -> ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = ( ( # ` { x } ) x. ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) )
12419, 22, 123mp2an 708 . . . . 5 |- ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = ( ( # ` { x } ) x. ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
125 vex 3203 . . . . . . 7 |- x e. _V
126 hashsng 13159 . . . . . . 7 |- ( x e. _V -> ( # ` { x } ) = 1 )
127125, 126ax-mp 5 . . . . . 6 |- ( # ` { x } ) = 1
128127oveq1i 6660 . . . . 5 |- ( ( # ` { x } ) x. ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = ( 1 x. ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
129 hashcl 13147 . . . . . . . 8 |- ( { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } e. Fin -> ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) e. NN0 )
13022, 129ax-mp 5 . . . . . . 7 |- ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) e. NN0
131130nn0cni 11304 . . . . . 6 |- ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) e. CC
132131mulid2i 10043 . . . . 5 |- ( 1 x. ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } )
133124, 128, 1323eqtri 2648 . . . 4 |- ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = ( # ` { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } )
134122, 133syl6breqr 4695 . . 3 |- ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) -> 2 || ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) )
13516, 18, 28, 134fsumdvds 15030 . 2 |- ( ph -> 2 || sum_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) )
1364, 13, 15mp2an 708 . . . . . 6 |- ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin
137 xpfi 8231 . . . . . 6 |- ( ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin /\ ( 0 ... N ) e. Fin ) -> ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin )
138136, 20, 137mp2an 708 . . . . 5 |- ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin
139 rabfi 8185 . . . . 5 |- ( ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin -> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } e. Fin )
140138, 139ax-mp 5 . . . 4 |- { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } e. Fin
14129nncnd 11036 . . . . . . . . . . . 12 |- ( ph -> N e. CC )
142 npcan1 10455 . . . . . . . . . . . 12 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
143141, 142syl 17 . . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
144 nnm1nn0 11334 . . . . . . . . . . . . . 14 |- ( N e. NN -> ( N - 1 ) e. NN0 )
14529, 144syl 17 . . . . . . . . . . . . 13 |- ( ph -> ( N - 1 ) e. NN0 )
146145nn0zd 11480 . . . . . . . . . . . 12 |- ( ph -> ( N - 1 ) e. ZZ )
147 uzid 11702 . . . . . . . . . . . 12 |- ( ( N - 1 ) e. ZZ -> ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
148 peano2uz 11741 . . . . . . . . . . . 12 |- ( ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
149146, 147, 1483syl 18 . . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
150143, 149eqeltrrd 2702 . . . . . . . . . 10 |- ( ph -> N e. ( ZZ>= ` ( N - 1 ) ) )
151 fzss2 12381 . . . . . . . . . 10 |- ( N e. ( ZZ>= ` ( N - 1 ) ) -> ( 0 ... ( N - 1 ) ) C_ ( 0 ... N ) )
152 ssralv 3666 . . . . . . . . . 10 |- ( ( 0 ... ( N - 1 ) ) C_ ( 0 ... N ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
153150, 151, 1523syl 18 . . . . . . . . 9 |- ( ph -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
154153adantr 481 . . . . . . . 8 |- ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
155 raldifb 3750 . . . . . . . . . . . 12 |- ( A. j e. ( 0 ... N ) ( j e/ { ( 2nd ` t ) } -> -. i = [_ ( 1st ` t ) / s ]_ C ) <-> A. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) -. i = [_ ( 1st ` t ) / s ]_ C )
156 nfv 1843 . . . . . . . . . . . . . . 15 |- F/ j ph
157 nfcsb1v 3549 . . . . . . . . . . . . . . . 16 |- F/_ j [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C
158157nfeq2 2780 . . . . . . . . . . . . . . 15 |- F/ j N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C
159156, 158nfan 1828 . . . . . . . . . . . . . 14 |- F/ j ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
160 nfv 1843 . . . . . . . . . . . . . 14 |- F/ j i e. ( 0 ... ( N - 1 ) )
161159, 160nfan 1828 . . . . . . . . . . . . 13 |- F/ j ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) )
162 nnel 2906 . . . . . . . . . . . . . . . . 17 |- ( -. j e/ { ( 2nd ` t ) } <-> j e. { ( 2nd ` t ) } )
163 velsn 4193 . . . . . . . . . . . . . . . . 17 |- ( j e. { ( 2nd ` t ) } <-> j = ( 2nd ` t ) )
164162, 163bitri 264 . . . . . . . . . . . . . . . 16 |- ( -. j e/ { ( 2nd ` t ) } <-> j = ( 2nd ` t ) )
165 csbeq1a 3542 . . . . . . . . . . . . . . . . . . . . 21 |- ( j = ( 2nd ` t ) -> [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
166165eqeq2d 2632 . . . . . . . . . . . . . . . . . . . 20 |- ( j = ( 2nd ` t ) -> ( N = [_ ( 1st ` t ) / s ]_ C <-> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
167166biimparc 504 . . . . . . . . . . . . . . . . . . 19 |- ( ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ j = ( 2nd ` t ) ) -> N = [_ ( 1st ` t ) / s ]_ C )
16829nnred 11035 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ph -> N e. RR )
169168ltm1d 10956 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ph -> ( N - 1 ) < N )
170145nn0red 11352 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ph -> ( N - 1 ) e. RR )
171170, 168ltnled 10184 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ph -> ( ( N - 1 ) < N <-> -. N <_ ( N - 1 ) ) )
172169, 171mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ph -> -. N <_ ( N - 1 ) )
173 elfzle2 12345 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N e. ( 0 ... ( N - 1 ) ) -> N <_ ( N - 1 ) )
174172, 173nsyl 135 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> -. N e. ( 0 ... ( N - 1 ) ) )
175 eleq1 2689 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = N -> ( i e. ( 0 ... ( N - 1 ) ) <-> N e. ( 0 ... ( N - 1 ) ) ) )
176175notbid 308 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = N -> ( -. i e. ( 0 ... ( N - 1 ) ) <-> -. N e. ( 0 ... ( N - 1 ) ) ) )
177174, 176syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( i = N -> -. i e. ( 0 ... ( N - 1 ) ) ) )
178177con2d 129 . . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( i e. ( 0 ... ( N - 1 ) ) -> -. i = N ) )
179178imp 445 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ i e. ( 0 ... ( N - 1 ) ) ) -> -. i = N )
180 eqeq2 2633 . . . . . . . . . . . . . . . . . . . . 21 |- ( N = [_ ( 1st ` t ) / s ]_ C -> ( i = N <-> i = [_ ( 1st ` t ) / s ]_ C ) )
181180notbid 308 . . . . . . . . . . . . . . . . . . . 20 |- ( N = [_ ( 1st ` t ) / s ]_ C -> ( -. i = N <-> -. i = [_ ( 1st ` t ) / s ]_ C ) )
182179, 181syl5ibcom 235 . . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( N = [_ ( 1st ` t ) / s ]_ C -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
183167, 182syl5 34 . . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ j = ( 2nd ` t ) ) -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
184183expdimp 453 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ i e. ( 0 ... ( N - 1 ) ) ) /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) -> ( j = ( 2nd ` t ) -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
185184an32s 846 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( j = ( 2nd ` t ) -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
186164, 185syl5bi 232 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( -. j e/ { ( 2nd ` t ) } -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
187 idd 24 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( -. i = [_ ( 1st ` t ) / s ]_ C -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
188186, 187jad 174 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( ( j e/ { ( 2nd ` t ) } -> -. i = [_ ( 1st ` t ) / s ]_ C ) -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
189188adantr 481 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) /\ j e. ( 0 ... N ) ) -> ( ( j e/ { ( 2nd ` t ) } -> -. i = [_ ( 1st ` t ) / s ]_ C ) -> -. i = [_ ( 1st ` t ) / s ]_ C ) )
190161, 189ralimdaa 2958 . . . . . . . . . . . 12 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( A. j e. ( 0 ... N ) ( j e/ { ( 2nd ` t ) } -> -. i = [_ ( 1st ` t ) / s ]_ C ) -> A. j e. ( 0 ... N ) -. i = [_ ( 1st ` t ) / s ]_ C ) )
191155, 190syl5bir 233 . . . . . . . . . . 11 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( A. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) -. i = [_ ( 1st ` t ) / s ]_ C -> A. j e. ( 0 ... N ) -. i = [_ ( 1st ` t ) / s ]_ C ) )
192191con3d 148 . . . . . . . . . 10 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( -. A. j e. ( 0 ... N ) -. i = [_ ( 1st ` t ) / s ]_ C -> -. A. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) -. i = [_ ( 1st ` t ) / s ]_ C ) )
193 dfrex2 2996 . . . . . . . . . 10 |- ( E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> -. A. j e. ( 0 ... N ) -. i = [_ ( 1st ` t ) / s ]_ C )
194 dfrex2 2996 . . . . . . . . . 10 |- ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> -. A. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) -. i = [_ ( 1st ` t ) / s ]_ C )
195192, 193, 1943imtr4g 285 . . . . . . . . 9 |- ( ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) /\ i e. ( 0 ... ( N - 1 ) ) ) -> ( E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C -> E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
196195ralimdva 2962 . . . . . . . 8 |- ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
197154, 196syld 47 . . . . . . 7 |- ( ( ph /\ N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
198197expimpd 629 . . . . . 6 |- ( ph -> ( ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
199198adantr 481 . . . . 5 |- ( ( ph /\ t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
200199ss2rabdv 3683 . . . 4 |- ( ph -> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } C_ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } )
201 hashssdif 13200 . . . 4 |- ( ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } e. Fin /\ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } C_ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) -> ( # ` ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) = ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) )
202140, 200, 201sylancr 695 . . 3 |- ( ph -> ( # ` ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) = ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) )
203 xp2nd 7199 . . . . . . . 8 |- ( t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` t ) e. ( 0 ... N ) )
204 df-ne 2795 . . . . . . . . . . . 12 |- ( N =/= [_ ( 1st ` t ) / s ]_ C <-> -. N = [_ ( 1st ` t ) / s ]_ C )
205204ralbii 2980 . . . . . . . . . . 11 |- ( A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C <-> A. j e. ( 0 ... N ) -. N = [_ ( 1st ` t ) / s ]_ C )
206 ralnex 2992 . . . . . . . . . . 11 |- ( A. j e. ( 0 ... N ) -. N = [_ ( 1st ` t ) / s ]_ C <-> -. E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C )
207205, 206bitri 264 . . . . . . . . . 10 |- ( A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C <-> -. E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C )
20829nnnn0d 11351 . . . . . . . . . . . . . . . . . . 19 |- ( ph -> N e. NN0 )
209 nn0uz 11722 . . . . . . . . . . . . . . . . . . 19 |- NN0 = ( ZZ>= ` 0 )
210208, 209syl6eleq 2711 . . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. ( ZZ>= ` 0 ) )
211143, 210eqeltrd 2701 . . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` 0 ) )
212 fzsplit2 12366 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( N - 1 ) + 1 ) e. ( ZZ>= ` 0 ) /\ N e. ( ZZ>= ` ( N - 1 ) ) ) -> ( 0 ... N ) = ( ( 0 ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) )
213211, 150, 212syl2anc 693 . . . . . . . . . . . . . . . 16 |- ( ph -> ( 0 ... N ) = ( ( 0 ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) )
214143oveq1d 6665 . . . . . . . . . . . . . . . . . 18 |- ( ph -> ( ( ( N - 1 ) + 1 ) ... N ) = ( N ... N ) )
21529nnzd 11481 . . . . . . . . . . . . . . . . . . 19 |- ( ph -> N e. ZZ )
216 fzsn 12383 . . . . . . . . . . . . . . . . . . 19 |- ( N e. ZZ -> ( N ... N ) = { N } )
217215, 216syl 17 . . . . . . . . . . . . . . . . . 18 |- ( ph -> ( N ... N ) = { N } )
218214, 217eqtrd 2656 . . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( ( N - 1 ) + 1 ) ... N ) = { N } )
219218uneq2d 3767 . . . . . . . . . . . . . . . 16 |- ( ph -> ( ( 0 ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) = ( ( 0 ... ( N - 1 ) ) u. { N } ) )
220213, 219eqtrd 2656 . . . . . . . . . . . . . . 15 |- ( ph -> ( 0 ... N ) = ( ( 0 ... ( N - 1 ) ) u. { N } ) )
221220raleqdv 3144 . . . . . . . . . . . . . 14 |- ( ph -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. ( ( 0 ... ( N - 1 ) ) u. { N } ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
222 difss 3737 . . . . . . . . . . . . . . . . . 18 |- ( ( 0 ... N ) \ { ( 2nd ` t ) } ) C_ ( 0 ... N )
223 ssrexv 3667 . . . . . . . . . . . . . . . . . 18 |- ( ( ( 0 ... N ) \ { ( 2nd ` t ) } ) C_ ( 0 ... N ) -> ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
224222, 223ax-mp 5 . . . . . . . . . . . . . . . . 17 |- ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C )
225224ralimi 2952 . . . . . . . . . . . . . . . 16 |- ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C )
226225biantrurd 529 . . . . . . . . . . . . . . 15 |- ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> ( A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C /\ A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
227 ralunb 3794 . . . . . . . . . . . . . . 15 |- ( A. i e. ( ( 0 ... ( N - 1 ) ) u. { N } ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C /\ A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
228226, 227syl6rbbr 279 . . . . . . . . . . . . . 14 |- ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> ( A. i e. ( ( 0 ... ( N - 1 ) ) u. { N } ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
229221, 228sylan9bb 736 . . . . . . . . . . . . 13 |- ( ( ph /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
230229adantlr 751 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) )
231 nn0fz0 12437 . . . . . . . . . . . . . . . 16 |- ( N e. NN0 <-> N e. ( 0 ... N ) )
232208, 231sylib 208 . . . . . . . . . . . . . . 15 |- ( ph -> N e. ( 0 ... N ) )
233232ad2antrr 762 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> N e. ( 0 ... N ) )
234 eqeq1 2626 . . . . . . . . . . . . . . . . 17 |- ( i = N -> ( i = [_ ( 1st ` t ) / s ]_ C <-> N = [_ ( 1st ` t ) / s ]_ C ) )
235234rexbidv 3052 . . . . . . . . . . . . . . . 16 |- ( i = N -> ( E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C ) )
236235rspcva 3307 . . . . . . . . . . . . . . 15 |- ( ( N e. ( 0 ... N ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) -> E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C )
237 nfv 1843 . . . . . . . . . . . . . . . . 17 |- F/ j ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) )
238 nfcv 2764 . . . . . . . . . . . . . . . . . 18 |- F/_ j ( 0 ... ( N - 1 ) )
239 nfre1 3005 . . . . . . . . . . . . . . . . . 18 |- F/ j E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C
240238, 239nfral 2945 . . . . . . . . . . . . . . . . 17 |- F/ j A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C
241237, 240nfan 1828 . . . . . . . . . . . . . . . 16 |- F/ j ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C )
242 eleq1 2689 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N = [_ ( 1st ` t ) / s ]_ C -> ( N e. ( 0 ... ( N - 1 ) ) <-> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
243242notbid 308 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( N = [_ ( 1st ` t ) / s ]_ C -> ( -. N e. ( 0 ... ( N - 1 ) ) <-> -. [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
244174, 243syl5ibcom 235 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( N = [_ ( 1st ` t ) / s ]_ C -> -. [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
245244ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) -> ( N = [_ ( 1st ` t ) / s ]_ C -> -. [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
246 eldifsn 4317 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) <-> ( j e. ( 0 ... N ) /\ j =/= ( 2nd ` t ) ) )
247 diffi 8192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( 0 ... N ) e. Fin -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) e. Fin )
24820, 247ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( 0 ... N ) \ { ( 2nd ` t ) } ) e. Fin
249 ssrab2 3687 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } C_ ( ( 0 ... N ) \ { ( 2nd ` t ) } )
250 ssdomg 8001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( ( 0 ... N ) \ { ( 2nd ` t ) } ) e. Fin -> ( { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } C_ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) -> { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ~<_ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) )
251248, 249, 250mp2 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ~<_ ( ( 0 ... N ) \ { ( 2nd ` t ) } )
252 hashdifsn 13202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( ( 0 ... N ) e. Fin /\ ( 2nd ` t ) e. ( 0 ... N ) ) -> ( # ` ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) = ( ( # ` ( 0 ... N ) ) - 1 ) )
25320, 252mpan 706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( 2nd ` t ) e. ( 0 ... N ) -> ( # ` ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) = ( ( # ` ( 0 ... N ) ) - 1 ) )
254 1cnd 10056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ph -> 1 e. CC )
255141, 254, 254addsubd 10413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ph -> ( ( N + 1 ) - 1 ) = ( ( N - 1 ) + 1 ) )
256 hashfz0 13219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( N e. NN0 -> ( # ` ( 0 ... N ) ) = ( N + 1 ) )
257208, 256syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ph -> ( # ` ( 0 ... N ) ) = ( N + 1 ) )
258257oveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ph -> ( ( # ` ( 0 ... N ) ) - 1 ) = ( ( N + 1 ) - 1 ) )
259 hashfz0 13219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( N - 1 ) e. NN0 -> ( # ` ( 0 ... ( N - 1 ) ) ) = ( ( N - 1 ) + 1 ) )
260145, 259syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ph -> ( # ` ( 0 ... ( N - 1 ) ) ) = ( ( N - 1 ) + 1 ) )
261255, 258, 2603eqtr4d 2666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ph -> ( ( # ` ( 0 ... N ) ) - 1 ) = ( # ` ( 0 ... ( N - 1 ) ) ) )
262253, 261sylan9eqr 2678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) -> ( # ` ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) = ( # ` ( 0 ... ( N - 1 ) ) ) )
263 fzfi 12771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( 0 ... ( N - 1 ) ) e. Fin
264 hashen 13135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( ( ( 0 ... N ) \ { ( 2nd ` t ) } ) e. Fin /\ ( 0 ... ( N - 1 ) ) e. Fin ) -> ( ( # ` ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) = ( # ` ( 0 ... ( N - 1 ) ) ) <-> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ~~ ( 0 ... ( N - 1 ) ) ) )
265248, 263, 264mp2an 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( # ` ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) = ( # ` ( 0 ... ( N - 1 ) ) ) <-> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ~~ ( 0 ... ( N - 1 ) ) )
266262, 265sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ~~ ( 0 ... ( N - 1 ) ) )
267 rabfi 8185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( ( 0 ... N ) \ { ( 2nd ` t ) } ) e. Fin -> { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } e. Fin )
268248, 267ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } e. Fin
269 eleq1 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ( i = [_ ( 1st ` t ) / s ]_ C -> ( i e. ( 0 ... ( N - 1 ) ) <-> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
270269biimpac 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( ( i e. ( 0 ... ( N - 1 ) ) /\ i = [_ ( 1st ` t ) / s ]_ C ) -> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) )
271 rabid 3116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } <-> ( j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) /\ [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
272271simplbi2com 657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) -> ( j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) -> j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ) )
273270, 272syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ( ( i e. ( 0 ... ( N - 1 ) ) /\ i = [_ ( 1st ` t ) / s ]_ C ) -> ( j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) -> j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ) )
274273impancom 456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ( ( i e. ( 0 ... ( N - 1 ) ) /\ j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) -> ( i = [_ ( 1st ` t ) / s ]_ C -> j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ) )
275274ancrd 577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ( ( i e. ( 0 ... ( N - 1 ) ) /\ j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) -> ( i = [_ ( 1st ` t ) / s ]_ C -> ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) ) )
276275expimpd 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( i e. ( 0 ... ( N - 1 ) ) -> ( ( j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) /\ i = [_ ( 1st ` t ) / s ]_ C ) -> ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) ) )
277276reximdv2 3014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( i e. ( 0 ... ( N - 1 ) ) -> ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> E. j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } i = [_ ( 1st ` t ) / s ]_ C ) )
278271simplbi 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } -> j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) )
279274pm4.71rd 667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ( ( i e. ( 0 ... ( N - 1 ) ) /\ j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) -> ( i = [_ ( 1st ` t ) / s ]_ C <-> ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) ) )
280 df-mpt 4730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) = { <. k , i >. | ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) }
281 nfv 1843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- F/ k ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C )
282 nfrab1 3122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- F/_ j { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) }
283282nfcri 2758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- F/ j k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) }
284 nfcsb1v 3549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- F/_ j [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C
285284nfeq2 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- F/ j i = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C
286283, 285nfan 1828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- F/ j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C )
287 eleq1 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- ( j = k -> ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } <-> k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ) )
288 csbeq1a 3542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- ( j = k -> [_ ( 1st ` t ) / s ]_ C = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C )
289288eqeq2d 2632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- ( j = k -> ( i = [_ ( 1st ` t ) / s ]_ C <-> i = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
290287, 289anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- ( j = k -> ( ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) <-> ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) ) )
291281, 286, 290cbvopab1 4723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- { <. j , i >. | ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) } = { <. k , i >. | ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) }
292280, 291eqtr4i 2647 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) = { <. j , i >. | ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) }
293292breqi 4659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i <-> j { <. j , i >. | ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) } i )
294 df-br 4654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( j { <. j , i >. | ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) } i <-> <. j , i >. e. { <. j , i >. | ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) } )
295 opabid 4982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( <. j , i >. e. { <. j , i >. | ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) } <-> ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) )
296293, 294, 2953bitri 286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ( j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i <-> ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } /\ i = [_ ( 1st ` t ) / s ]_ C ) )
297279, 296syl6bbr 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ( ( i e. ( 0 ... ( N - 1 ) ) /\ j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) -> ( i = [_ ( 1st ` t ) / s ]_ C <-> j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
298278, 297sylan2 491 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ( ( i e. ( 0 ... ( N - 1 ) ) /\ j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ) -> ( i = [_ ( 1st ` t ) / s ]_ C <-> j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
299298rexbidva 3049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( i e. ( 0 ... ( N - 1 ) ) -> ( E. j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
300 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- F/_ p { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) }
301 nfv 1843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- F/ p j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i
302 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- F/_ j p
303282, 284nfmpt 4746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- F/_ j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C )
304 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- F/_ j i
305302, 303, 304nfbr 4699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- F/ j p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i
306 breq1 4656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ( j = p -> ( j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i <-> p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
307282, 300, 301, 305, 306cbvrexf 3166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( E. j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } j ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i <-> E. p e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i )
308299, 307syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( i e. ( 0 ... ( N - 1 ) ) -> ( E. j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } i = [_ ( 1st ` t ) / s ]_ C <-> E. p e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
309277, 308sylibd 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( i e. ( 0 ... ( N - 1 ) ) -> ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> E. p e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
310309ralimia 2950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. p e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i )
311 eqid 2622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) = ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C )
312 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- F/_ j k
313 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- F/_ j ( ( 0 ... N ) \ { ( 2nd ` t ) } )
314284nfel1 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- F/ j [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) )
315288eleq1d 2686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( j = k -> ( [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) <-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
316312, 313, 314, 315elrabf 3360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } <-> ( k e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) /\ [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
317316simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } -> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) )
318311, 317fmpti 6383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) : { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } --> ( 0 ... ( N - 1 ) )
319310, 318jctil 560 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> ( ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) : { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } --> ( 0 ... ( N - 1 ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. p e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
320 dffo4 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) : { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } -onto-> ( 0 ... ( N - 1 ) ) <-> ( ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) : { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } --> ( 0 ... ( N - 1 ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. p e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } p ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) i ) )
321319, 320sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) : { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } -onto-> ( 0 ... ( N - 1 ) ) )
322 fodomfi 8239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } e. Fin /\ ( k e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } |-> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C ) : { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } -onto-> ( 0 ... ( N - 1 ) ) ) -> ( 0 ... ( N - 1 ) ) ~<_ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } )
323268, 321, 322sylancr 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> ( 0 ... ( N - 1 ) ) ~<_ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } )
324 endomtr 8014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ~~ ( 0 ... ( N - 1 ) ) /\ ( 0 ... ( N - 1 ) ) ~<_ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ) -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ~<_ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } )
325266, 323, 324syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ~<_ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } )
326 sbth 8080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ~<_ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) /\ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ~<_ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ) -> { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ~~ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) )
327251, 325, 326sylancr 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ~~ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) )
328 fisseneq 8171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( ( 0 ... N ) \ { ( 2nd ` t ) } ) e. Fin /\ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } C_ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) /\ { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } ~~ ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) -> { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } = ( ( 0 ... N ) \ { ( 2nd ` t ) } ) )
329248, 249, 327, 328mp3an12i 1428 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } = ( ( 0 ... N ) \ { ( 2nd ` t ) } ) )
330329eleq2d 2687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } <-> j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) )
331330biimpar 502 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) -> j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } )
332288equcoms 1947 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( k = j -> [_ ( 1st ` t ) / s ]_ C = [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C )
333332eqcomd 2628 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( k = j -> [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C = [_ ( 1st ` t ) / s ]_ C )
334333eleq1d 2686 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( k = j -> ( [_ k / j ]_ [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) <-> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
335334, 317vtoclga 3272 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( j e. { j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) | [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) } -> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) )
336331, 335syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) ) -> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) )
337246, 336sylan2br 493 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ ( j e. ( 0 ... N ) /\ j =/= ( 2nd ` t ) ) ) -> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) )
338337expr 643 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) -> ( j =/= ( 2nd ` t ) -> [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) ) )
339338necon1bd 2812 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) -> ( -. [_ ( 1st ` t ) / s ]_ C e. ( 0 ... ( N - 1 ) ) -> j = ( 2nd ` t ) ) )
340245, 339syld 47 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) -> ( N = [_ ( 1st ` t ) / s ]_ C -> j = ( 2nd ` t ) ) )
341340imp 445 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) /\ N = [_ ( 1st ` t ) / s ]_ C ) -> j = ( 2nd ` t ) )
342341, 165syl 17 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) /\ N = [_ ( 1st ` t ) / s ]_ C ) -> [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
343 eqtr 2641 . . . . . . . . . . . . . . . . . . . 20 |- ( ( N = [_ ( 1st ` t ) / s ]_ C /\ [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
344343ex 450 . . . . . . . . . . . . . . . . . . 19 |- ( N = [_ ( 1st ` t ) / s ]_ C -> ( [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
345344adantl 482 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) /\ N = [_ ( 1st ` t ) / s ]_ C ) -> ( [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
346342, 345mpd 15 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) /\ j e. ( 0 ... N ) ) /\ N = [_ ( 1st ` t ) / s ]_ C ) -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
347346exp31 630 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( j e. ( 0 ... N ) -> ( N = [_ ( 1st ` t ) / s ]_ C -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) ) )
348241, 158, 347rexlimd 3026 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
349236, 348syl5 34 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( ( N e. ( 0 ... N ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
350233, 349mpand 711 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
351350pm4.71rd 667 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
352235ralsng 4218 . . . . . . . . . . . . . 14 |- ( N e. NN -> ( A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C ) )
35329, 352syl 17 . . . . . . . . . . . . 13 |- ( ph -> ( A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C ) )
354353ad2antrr 762 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( A. i e. { N } E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C ) )
355230, 351, 3543bitr3rd 299 . . . . . . . . . . 11 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C <-> ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
356355notbid 308 . . . . . . . . . 10 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( -. E. j e. ( 0 ... N ) N = [_ ( 1st ` t ) / s ]_ C <-> -. ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
357207, 356syl5bb 272 . . . . . . . . 9 |- ( ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C <-> -. ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
358357pm5.32da 673 . . . . . . . 8 |- ( ( ph /\ ( 2nd ` t ) e. ( 0 ... N ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) ) )
359203, 358sylan2 491 . . . . . . 7 |- ( ( ph /\ t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) ) )
360359rabbidva 3188 . . . . . 6 |- ( ph -> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C ) } = { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) } )
361 nfv 1843 . . . . . . . . . . . 12 |- F/ y t = <. x , k >.
362 nfv 1843 . . . . . . . . . . . . 13 |- F/ y x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
363 nfrab1 3122 . . . . . . . . . . . . . 14 |- F/_ y { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) }
364363nfcri 2758 . . . . . . . . . . . . 13 |- F/ y k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) }
365362, 364nfan 1828 . . . . . . . . . . . 12 |- F/ y ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } )
366361, 365nfan 1828 . . . . . . . . . . 11 |- F/ y ( t = <. x , k >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
367 nfv 1843 . . . . . . . . . . 11 |- F/ k ( t = <. x , y >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) )
368 opeq2 4403 . . . . . . . . . . . . 13 |- ( k = y -> <. x , k >. = <. x , y >. )
369368eqeq2d 2632 . . . . . . . . . . . 12 |- ( k = y -> ( t = <. x , k >. <-> t = <. x , y >. ) )
370 eleq1 2689 . . . . . . . . . . . . . . 15 |- ( k = y -> ( k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } <-> y e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
371 rabid 3116 . . . . . . . . . . . . . . 15 |- ( y e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } <-> ( y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) )
372370, 371syl6bb 276 . . . . . . . . . . . . . 14 |- ( k = y -> ( k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } <-> ( y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) )
373372anbi2d 740 . . . . . . . . . . . . 13 |- ( k = y -> ( ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) <-> ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) ) )
374 3anass 1042 . . . . . . . . . . . . 13 |- ( ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) <-> ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) )
375373, 374syl6bbr 278 . . . . . . . . . . . 12 |- ( k = y -> ( ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) <-> ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) )
376369, 375anbi12d 747 . . . . . . . . . . 11 |- ( k = y -> ( ( t = <. x , k >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) <-> ( t = <. x , y >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) ) )
377366, 367, 376cbvex 2272 . . . . . . . . . 10 |- ( E. k ( t = <. x , k >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) <-> E. y ( t = <. x , y >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) )
378377exbii 1774 . . . . . . . . 9 |- ( E. x E. k ( t = <. x , k >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) <-> E. x E. y ( t = <. x , y >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) )
379 eliunxp 5259 . . . . . . . . 9 |- ( t e. U_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) <-> E. x E. k ( t = <. x , k >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ k e. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) )
380 elopab 4983 . . . . . . . . 9 |- ( t e. { <. x , y >. | ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) } <-> E. x E. y ( t = <. x , y >. /\ ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) ) )
381378, 379, 3803bitr4i 292 . . . . . . . 8 |- ( t e. U_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) <-> t e. { <. x , y >. | ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) } )
382381eqriv 2619 . . . . . . 7 |- U_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) = { <. x , y >. | ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) }
383 vex 3203 . . . . . . . . . . . . . 14 |- y e. _V
384125, 383op2ndd 7179 . . . . . . . . . . . . 13 |- ( t = <. x , y >. -> ( 2nd ` t ) = y )
385384sneqd 4189 . . . . . . . . . . . 12 |- ( t = <. x , y >. -> { ( 2nd ` t ) } = { y } )
386385difeq2d 3728 . . . . . . . . . . 11 |- ( t = <. x , y >. -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) = ( ( 0 ... N ) \ { y } ) )
387125, 383op1std 7178 . . . . . . . . . . . . 13 |- ( t = <. x , y >. -> ( 1st ` t ) = x )
388387csbeq1d 3540 . . . . . . . . . . . 12 |- ( t = <. x , y >. -> [_ ( 1st ` t ) / s ]_ C = [_ x / s ]_ C )
389388eqeq2d 2632 . . . . . . . . . . 11 |- ( t = <. x , y >. -> ( i = [_ ( 1st ` t ) / s ]_ C <-> i = [_ x / s ]_ C ) )
390386, 389rexeqbidv 3153 . . . . . . . . . 10 |- ( t = <. x , y >. -> ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C ) )
391390ralbidv 2986 . . . . . . . . 9 |- ( t = <. x , y >. -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C ) )
392388neeq2d 2854 . . . . . . . . . 10 |- ( t = <. x , y >. -> ( N =/= [_ ( 1st ` t ) / s ]_ C <-> N =/= [_ x / s ]_ C ) )
393392ralbidv 2986 . . . . . . . . 9 |- ( t = <. x , y >. -> ( A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C <-> A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) )
394391, 393anbi12d 747 . . . . . . . 8 |- ( t = <. x , y >. -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) )
395394rabxp 5154 . . . . . . 7 |- { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C ) } = { <. x , y >. | ( x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ y e. ( 0 ... N ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) ) }
396382, 395eqtr4i 2647 . . . . . 6 |- U_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) = { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ ( 1st ` t ) / s ]_ C ) }
397 difrab 3901 . . . . . 6 |- ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) ) }
398360, 396, 3973eqtr4g 2681 . . . . 5 |- ( ph -> U_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) = ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) )
399398fveq2d 6195 . . . 4 |- ( ph -> ( # ` U_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = ( # ` ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) )
40024a1i 11 . . . . 5 |- ( ( ph /\ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) -> ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) e. Fin )
401 inxp 5254 . . . . . . . . . 10 |- ( ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) i^i ( { t } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = ( ( { x } i^i { t } ) X. ( { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } i^i { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) )
402 df-ne 2795 . . . . . . . . . . . . 13 |- ( x =/= t <-> -. x = t )
403 disjsn2 4247 . . . . . . . . . . . . 13 |- ( x =/= t -> ( { x } i^i { t } ) = (/) )
404402, 403sylbir 225 . . . . . . . . . . . 12 |- ( -. x = t -> ( { x } i^i { t } ) = (/) )
405404xpeq1d 5138 . . . . . . . . . . 11 |- ( -. x = t -> ( ( { x } i^i { t } ) X. ( { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } i^i { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = ( (/) X. ( { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } i^i { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) )
406 0xp 5199 . . . . . . . . . . 11 |- ( (/) X. ( { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } i^i { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = (/)
407405, 406syl6eq 2672 . . . . . . . . . 10 |- ( -. x = t -> ( ( { x } i^i { t } ) X. ( { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } i^i { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = (/) )
408401, 407syl5eq 2668 . . . . . . . . 9 |- ( -. x = t -> ( ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) i^i ( { t } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = (/) )
409408orri 391 . . . . . . . 8 |- ( x = t \/ ( ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) i^i ( { t } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = (/) )
410409rgen2w 2925 . . . . . . 7 |- A. x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) A. t e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( x = t \/ ( ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) i^i ( { t } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = (/) )
411 sneq 4187 . . . . . . . . 9 |- ( x = t -> { x } = { t } )
412 csbeq1 3536 . . . . . . . . . . . . . 14 |- ( x = t -> [_ x / s ]_ C = [_ t / s ]_ C )
413412eqeq2d 2632 . . . . . . . . . . . . 13 |- ( x = t -> ( i = [_ x / s ]_ C <-> i = [_ t / s ]_ C ) )
414413rexbidv 3052 . . . . . . . . . . . 12 |- ( x = t -> ( E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C <-> E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C ) )
415414ralbidv 2986 . . . . . . . . . . 11 |- ( x = t -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C ) )
416412neeq2d 2854 . . . . . . . . . . . 12 |- ( x = t -> ( N =/= [_ x / s ]_ C <-> N =/= [_ t / s ]_ C ) )
417416ralbidv 2986 . . . . . . . . . . 11 |- ( x = t -> ( A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C <-> A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) )
418415, 417anbi12d 747 . . . . . . . . . 10 |- ( x = t -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) ) )
419418rabbidv 3189 . . . . . . . . 9 |- ( x = t -> { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } = { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } )
420411, 419xpeq12d 5140 . . . . . . . 8 |- ( x = t -> ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) = ( { t } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) )
421420disjor 4634 . . . . . . 7 |- (Disj x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) <-> A. x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) A. t e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( x = t \/ ( ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) i^i ( { t } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ t / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ t / s ]_ C ) } ) ) = (/) ) )
422410, 421mpbir 221 . . . . . 6 |- Disj x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } )
423422a1i 11 . . . . 5 |- ( ph -> Disj x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) )
42416, 400, 423hashiun 14554 . . . 4 |- ( ph -> ( # ` U_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = sum_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) )
425399, 424eqtr3d 2658 . . 3 |- ( ph -> ( # ` ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) = sum_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) )
426 fo1st 7188 . . . . . . . . . . . 12 |- 1st : _V -onto-> _V
427 fofun 6116 . . . . . . . . . . . 12 |- ( 1st : _V -onto-> _V -> Fun 1st )
428426, 427ax-mp 5 . . . . . . . . . . 11 |- Fun 1st
429 ssv 3625 . . . . . . . . . . . 12 |- { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } C_ _V
430 fof 6115 . . . . . . . . . . . . . 14 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
431426, 430ax-mp 5 . . . . . . . . . . . . 13 |- 1st : _V --> _V
432431fdmi 6052 . . . . . . . . . . . 12 |- dom 1st = _V
433429, 432sseqtr4i 3638 . . . . . . . . . . 11 |- { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } C_ dom 1st
434 fores 6124 . . . . . . . . . . 11 |- ( ( Fun 1st /\ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } C_ dom 1st ) -> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) )
435428, 433, 434mp2an 708 . . . . . . . . . 10 |- ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } )
436 fveq2 6191 . . . . . . . . . . . . . . . . . . 19 |- ( t = x -> ( 2nd ` t ) = ( 2nd ` x ) )
437436csbeq1d 3540 . . . . . . . . . . . . . . . . . 18 |- ( t = x -> [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
438 fveq2 6191 . . . . . . . . . . . . . . . . . . . 20 |- ( t = x -> ( 1st ` t ) = ( 1st ` x ) )
439438csbeq1d 3540 . . . . . . . . . . . . . . . . . . 19 |- ( t = x -> [_ ( 1st ` t ) / s ]_ C = [_ ( 1st ` x ) / s ]_ C )
440439csbeq2dv 3992 . . . . . . . . . . . . . . . . . 18 |- ( t = x -> [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C )
441437, 440eqtrd 2656 . . . . . . . . . . . . . . . . 17 |- ( t = x -> [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C )
442441eqeq2d 2632 . . . . . . . . . . . . . . . 16 |- ( t = x -> ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C <-> N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) )
443439eqeq2d 2632 . . . . . . . . . . . . . . . . . 18 |- ( t = x -> ( i = [_ ( 1st ` t ) / s ]_ C <-> i = [_ ( 1st ` x ) / s ]_ C ) )
444443rexbidv 3052 . . . . . . . . . . . . . . . . 17 |- ( t = x -> ( E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) )
445444ralbidv 2986 . . . . . . . . . . . . . . . 16 |- ( t = x -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) )
446442, 445anbi12d 747 . . . . . . . . . . . . . . 15 |- ( t = x -> ( ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) <-> ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) )
447446rexrab 3370 . . . . . . . . . . . . . 14 |- ( E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = s <-> E. x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) )
448 xp1st 7198 . . . . . . . . . . . . . . . . . . . 20 |- ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` x ) e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
449448anim1i 592 . . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) -> ( ( 1st ` x ) e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) )
450 eleq1 2689 . . . . . . . . . . . . . . . . . . . 20 |- ( ( 1st ` x ) = s -> ( ( 1st ` x ) e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) <-> s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) )
451 csbeq1a 3542 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( s = ( 1st ` x ) -> C = [_ ( 1st ` x ) / s ]_ C )
452451eqcoms 2630 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( 1st ` x ) = s -> C = [_ ( 1st ` x ) / s ]_ C )
453452eqcomd 2628 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( 1st ` x ) = s -> [_ ( 1st ` x ) / s ]_ C = C )
454453eqeq2d 2632 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 1st ` x ) = s -> ( i = [_ ( 1st ` x ) / s ]_ C <-> i = C ) )
455454rexbidv 3052 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1st ` x ) = s -> ( E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C <-> E. j e. ( 0 ... N ) i = C ) )
456455ralbidv 2986 . . . . . . . . . . . . . . . . . . . 20 |- ( ( 1st ` x ) = s -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C <-> A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) )
457450, 456anbi12d 747 . . . . . . . . . . . . . . . . . . 19 |- ( ( 1st ` x ) = s -> ( ( ( 1st ` x ) e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) <-> ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) )
458449, 457syl5ibcom 235 . . . . . . . . . . . . . . . . . 18 |- ( ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) -> ( ( 1st ` x ) = s -> ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) )
459458adantrl 752 . . . . . . . . . . . . . . . . 17 |- ( ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) -> ( ( 1st ` x ) = s -> ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) )
460459expimpd 629 . . . . . . . . . . . . . . . 16 |- ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) -> ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) )
461460rexlimiv 3027 . . . . . . . . . . . . . . 15 |- ( E. x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) -> ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) )
462 simplr 792 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
463 ovex 6678 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 0 ... N ) e. _V
464463enref 7988 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( 0 ... N ) ~~ ( 0 ... N )
465 phpreu 33393 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( 0 ... N ) e. Fin /\ ( 0 ... N ) ~~ ( 0 ... N ) ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C <-> A. i e. ( 0 ... N ) E! j e. ( 0 ... N ) i = C ) )
46620, 464, 465mp2an 708 . . . . . . . . . . . . . . . . . . . . . 22 |- ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C <-> A. i e. ( 0 ... N ) E! j e. ( 0 ... N ) i = C )
467466biimpi 206 . . . . . . . . . . . . . . . . . . . . 21 |- ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C -> A. i e. ( 0 ... N ) E! j e. ( 0 ... N ) i = C )
468 eqeq1 2626 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = N -> ( i = C <-> N = C ) )
469468reubidv 3126 . . . . . . . . . . . . . . . . . . . . . 22 |- ( i = N -> ( E! j e. ( 0 ... N ) i = C <-> E! j e. ( 0 ... N ) N = C ) )
470469rspcva 3307 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( N e. ( 0 ... N ) /\ A. i e. ( 0 ... N ) E! j e. ( 0 ... N ) i = C ) -> E! j e. ( 0 ... N ) N = C )
471232, 467, 470syl2an 494 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> E! j e. ( 0 ... N ) N = C )
472 riotacl 6625 . . . . . . . . . . . . . . . . . . . 20 |- ( E! j e. ( 0 ... N ) N = C -> ( iota_ j e. ( 0 ... N ) N = C ) e. ( 0 ... N ) )
473471, 472syl 17 . . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> ( iota_ j e. ( 0 ... N ) N = C ) e. ( 0 ... N ) )
474473adantlr 751 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> ( iota_ j e. ( 0 ... N ) N = C ) e. ( 0 ... N ) )
475 opelxpi 5148 . . . . . . . . . . . . . . . . . 18 |- ( ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( iota_ j e. ( 0 ... N ) N = C ) e. ( 0 ... N ) ) -> <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
476462, 474, 475syl2anc 693 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
477 riotasbc 6626 . . . . . . . . . . . . . . . . . . . . . 22 |- ( E! j e. ( 0 ... N ) N = C -> [. ( iota_ j e. ( 0 ... N ) N = C ) / j ]. N = C )
478471, 477syl 17 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> [. ( iota_ j e. ( 0 ... N ) N = C ) / j ]. N = C )
479 riotaex 6615 . . . . . . . . . . . . . . . . . . . . . 22 |- ( iota_ j e. ( 0 ... N ) N = C ) e. _V
480 sbceq2g 3990 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( iota_ j e. ( 0 ... N ) N = C ) e. _V -> ( [. ( iota_ j e. ( 0 ... N ) N = C ) / j ]. N = C <-> N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C ) )
481479, 480ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 |- ( [. ( iota_ j e. ( 0 ... N ) N = C ) / j ]. N = C <-> N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C )
482478, 481sylib 208 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C )
483482expcom 451 . . . . . . . . . . . . . . . . . . 19 |- ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C -> ( ph -> N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C ) )
484483imdistanri 727 . . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> ( N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) )
485484adantlr 751 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> ( N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) )
486 vex 3203 . . . . . . . . . . . . . . . . . . . . . . . 24 |- s e. _V
487486, 479op2ndd 7179 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( 2nd ` x ) = ( iota_ j e. ( 0 ... N ) N = C ) )
488487csbeq1d 3540 . . . . . . . . . . . . . . . . . . . . . 22 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> [_ ( 2nd ` x ) / j ]_ C = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C )
489 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- F/_ j s
490 nfriota1 6618 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- F/_ j ( iota_ j e. ( 0 ... N ) N = C )
491489, 490nfop 4418 . . . . . . . . . . . . . . . . . . . . . . . 24 |- F/_ j <. s , ( iota_ j e. ( 0 ... N ) N = C ) >.
492491nfeq2 2780 . . . . . . . . . . . . . . . . . . . . . . 23 |- F/ j x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >.
493486, 479op1std 7178 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( 1st ` x ) = s )
494493eqcomd 2628 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> s = ( 1st ` x ) )
495494, 451syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> C = [_ ( 1st ` x ) / s ]_ C )
496492, 495csbeq2d 3991 . . . . . . . . . . . . . . . . . . . . . 22 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> [_ ( 2nd ` x ) / j ]_ C = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C )
497488, 496eqtr3d 2658 . . . . . . . . . . . . . . . . . . . . 21 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C )
498497eqeq2d 2632 . . . . . . . . . . . . . . . . . . . 20 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C <-> N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) )
499495eqeq2d 2632 . . . . . . . . . . . . . . . . . . . . . 22 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( i = C <-> i = [_ ( 1st ` x ) / s ]_ C ) )
500492, 499rexbid 3051 . . . . . . . . . . . . . . . . . . . . 21 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( E. j e. ( 0 ... N ) i = C <-> E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) )
501500ralbidv 2986 . . . . . . . . . . . . . . . . . . . 20 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C <-> A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) )
502498, 501anbi12d 747 . . . . . . . . . . . . . . . . . . 19 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( ( N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) <-> ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) )
503493biantrud 528 . . . . . . . . . . . . . . . . . . 19 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) <-> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) ) )
504502, 503bitr2d 269 . . . . . . . . . . . . . . . . . 18 |- ( x = <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. -> ( ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) <-> ( N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) )
505504rspcev 3309 . . . . . . . . . . . . . . . . 17 |- ( ( <. s , ( iota_ j e. ( 0 ... N ) N = C ) >. e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( N = [_ ( iota_ j e. ( 0 ... N ) N = C ) / j ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) -> E. x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) )
506476, 485, 505syl2anc 693 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> E. x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) )
507506expl 648 . . . . . . . . . . . . . . 15 |- ( ph -> ( ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) -> E. x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) ) )
508461, 507impbid2 216 . . . . . . . . . . . . . 14 |- ( ph -> ( E. x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) /\ ( 1st ` x ) = s ) <-> ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) )
509447, 508syl5bb 272 . . . . . . . . . . . . 13 |- ( ph -> ( E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = s <-> ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) ) )
510509abbidv 2741 . . . . . . . . . . . 12 |- ( ph -> { s | E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = s } = { s | ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) } )
511 dfimafn 6245 . . . . . . . . . . . . . 14 |- ( ( Fun 1st /\ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } C_ dom 1st ) -> ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = { y | E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = y } )
512428, 433, 511mp2an 708 . . . . . . . . . . . . 13 |- ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = { y | E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = y }
513 nfcv 2764 . . . . . . . . . . . . . . . . . . 19 |- F/_ s ( 2nd ` t )
514 nfcsb1v 3549 . . . . . . . . . . . . . . . . . . 19 |- F/_ s [_ ( 1st ` t ) / s ]_ C
515513, 514nfcsb 3551 . . . . . . . . . . . . . . . . . 18 |- F/_ s [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C
516515nfeq2 2780 . . . . . . . . . . . . . . . . 17 |- F/ s N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C
517 nfcv 2764 . . . . . . . . . . . . . . . . . 18 |- F/_ s ( 0 ... N )
518514nfeq2 2780 . . . . . . . . . . . . . . . . . . 19 |- F/ s i = [_ ( 1st ` t ) / s ]_ C
519517, 518nfrex 3007 . . . . . . . . . . . . . . . . . 18 |- F/ s E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C
520517, 519nfral 2945 . . . . . . . . . . . . . . . . 17 |- F/ s A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C
521516, 520nfan 1828 . . . . . . . . . . . . . . . 16 |- F/ s ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C )
522 nfcv 2764 . . . . . . . . . . . . . . . 16 |- F/_ s ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) )
523521, 522nfrab 3123 . . . . . . . . . . . . . . 15 |- F/_ s { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) }
524 nfv 1843 . . . . . . . . . . . . . . 15 |- F/ s ( 1st ` x ) = y
525523, 524nfrex 3007 . . . . . . . . . . . . . 14 |- F/ s E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = y
526 nfv 1843 . . . . . . . . . . . . . 14 |- F/ y E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = s
527 eqeq2 2633 . . . . . . . . . . . . . . 15 |- ( y = s -> ( ( 1st ` x ) = y <-> ( 1st ` x ) = s ) )
528527rexbidv 3052 . . . . . . . . . . . . . 14 |- ( y = s -> ( E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = y <-> E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = s ) )
529525, 526, 528cbvab 2746 . . . . . . . . . . . . 13 |- { y | E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = y } = { s | E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = s }
530512, 529eqtri 2644 . . . . . . . . . . . 12 |- ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = { s | E. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( 1st ` x ) = s }
531 df-rab 2921 . . . . . . . . . . . 12 |- { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } = { s | ( s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C ) }
532510, 530, 5313eqtr4g 2681 . . . . . . . . . . 11 |- ( ph -> ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
533 foeq3 6113 . . . . . . . . . . 11 |- ( ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } -> ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) <-> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) )
534532, 533syl 17 . . . . . . . . . 10 |- ( ph -> ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> ( 1st " { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) <-> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) )
535435, 534mpbii 223 . . . . . . . . 9 |- ( ph -> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
536 fof 6115 . . . . . . . . 9 |- ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } -> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } --> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
537535, 536syl 17 . . . . . . . 8 |- ( ph -> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } --> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
538 fvres 6207 . . . . . . . . . . . 12 |- ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -> ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` x ) = ( 1st ` x ) )
539 fvres 6207 . . . . . . . . . . . 12 |- ( y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -> ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` y ) = ( 1st ` y ) )
540538, 539eqeqan12d 2638 . . . . . . . . . . 11 |- ( ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) -> ( ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` y ) <-> ( 1st ` x ) = ( 1st ` y ) ) )
541540adantl 482 . . . . . . . . . 10 |- ( ( ph /\ ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) -> ( ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` y ) <-> ( 1st ` x ) = ( 1st ` y ) ) )
542446elrab 3363 . . . . . . . . . . . . . . 15 |- ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } <-> ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) )
543 xp2nd 7199 . . . . . . . . . . . . . . . 16 |- ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` x ) e. ( 0 ... N ) )
544543anim1i 592 . . . . . . . . . . . . . . 15 |- ( ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) -> ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) )
545542, 544sylbi 207 . . . . . . . . . . . . . 14 |- ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -> ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) )
546 simpl 473 . . . . . . . . . . . . . . . . . 18 |- ( ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
547546a1i 11 . . . . . . . . . . . . . . . . 17 |- ( t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) -> N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C ) )
548547ss2rabi 3684 . . . . . . . . . . . . . . . 16 |- { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } C_ { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C }
549548sseli 3599 . . . . . . . . . . . . . . 15 |- ( y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -> y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C } )
550 fveq2 6191 . . . . . . . . . . . . . . . . . . . 20 |- ( t = y -> ( 2nd ` t ) = ( 2nd ` y ) )
551550csbeq1d 3540 . . . . . . . . . . . . . . . . . . 19 |- ( t = y -> [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` t ) / s ]_ C )
552 fveq2 6191 . . . . . . . . . . . . . . . . . . . . 21 |- ( t = y -> ( 1st ` t ) = ( 1st ` y ) )
553552csbeq1d 3540 . . . . . . . . . . . . . . . . . . . 20 |- ( t = y -> [_ ( 1st ` t ) / s ]_ C = [_ ( 1st ` y ) / s ]_ C )
554553csbeq2dv 3992 . . . . . . . . . . . . . . . . . . 19 |- ( t = y -> [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C )
555551, 554eqtrd 2656 . . . . . . . . . . . . . . . . . 18 |- ( t = y -> [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C )
556555eqeq2d 2632 . . . . . . . . . . . . . . . . 17 |- ( t = y -> ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C <-> N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) )
557556elrab 3363 . . . . . . . . . . . . . . . 16 |- ( y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C } <-> ( y e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) )
558 xp2nd 7199 . . . . . . . . . . . . . . . . 17 |- ( y e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` y ) e. ( 0 ... N ) )
559558anim1i 592 . . . . . . . . . . . . . . . 16 |- ( ( y e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) -> ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) )
560557, 559sylbi 207 . . . . . . . . . . . . . . 15 |- ( y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C } -> ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) )
561549, 560syl 17 . . . . . . . . . . . . . 14 |- ( y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -> ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) )
562545, 561anim12i 590 . . . . . . . . . . . . 13 |- ( ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) -> ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) )
563 an4 865 . . . . . . . . . . . . . . 15 |- ( ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) <-> ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) )
564563anbi2i 730 . . . . . . . . . . . . . 14 |- ( ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) ) <-> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) ) )
565 anass 681 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) <-> ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) )
566 ancom 466 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) <-> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) ) )
567565, 566bitr3i 266 . . . . . . . . . . . . . . . 16 |- ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) <-> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) ) )
568567anbi1i 731 . . . . . . . . . . . . . . 15 |- ( ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) <-> ( ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) )
569 anass 681 . . . . . . . . . . . . . . 15 |- ( ( ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) <-> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) ) )
570568, 569bitri 264 . . . . . . . . . . . . . 14 |- ( ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) <-> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) ) )
571 anass 681 . . . . . . . . . . . . . 14 |- ( ( ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) <-> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) ) )
572564, 570, 5713bitr4i 292 . . . . . . . . . . . . 13 |- ( ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) ) /\ ( ( 2nd ` y ) e. ( 0 ... N ) /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) <-> ( ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) )
573562, 572sylib 208 . . . . . . . . . . . 12 |- ( ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) -> ( ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) )
574 phpreu 33393 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( 0 ... N ) e. Fin /\ ( 0 ... N ) ~~ ( 0 ... N ) ) -> ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C <-> A. i e. ( 0 ... N ) E! j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) )
57520, 464, 574mp2an 708 . . . . . . . . . . . . . . . . . . . 20 |- ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C <-> A. i e. ( 0 ... N ) E! j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C )
576 reurmo 3161 . . . . . . . . . . . . . . . . . . . . 21 |- ( E! j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C -> E* j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C )
577576ralimi 2952 . . . . . . . . . . . . . . . . . . . 20 |- ( A. i e. ( 0 ... N ) E! j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C -> A. i e. ( 0 ... N ) E* j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C )
578575, 577sylbi 207 . . . . . . . . . . . . . . . . . . 19 |- ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C -> A. i e. ( 0 ... N ) E* j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C )
579 eqeq1 2626 . . . . . . . . . . . . . . . . . . . . 21 |- ( i = N -> ( i = [_ ( 1st ` x ) / s ]_ C <-> N = [_ ( 1st ` x ) / s ]_ C ) )
580579rmobidv 3131 . . . . . . . . . . . . . . . . . . . 20 |- ( i = N -> ( E* j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C <-> E* j e. ( 0 ... N ) N = [_ ( 1st ` x ) / s ]_ C ) )
581580rspcva 3307 . . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ( 0 ... N ) /\ A. i e. ( 0 ... N ) E* j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) -> E* j e. ( 0 ... N ) N = [_ ( 1st ` x ) / s ]_ C )
582232, 578, 581syl2an 494 . . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) -> E* j e. ( 0 ... N ) N = [_ ( 1st ` x ) / s ]_ C )
583 nfv 1843 . . . . . . . . . . . . . . . . . . 19 |- F/ k N = [_ ( 1st ` x ) / s ]_ C
584583rmo3 3528 . . . . . . . . . . . . . . . . . 18 |- ( E* j e. ( 0 ... N ) N = [_ ( 1st ` x ) / s ]_ C <-> A. j e. ( 0 ... N ) A. k e. ( 0 ... N ) ( ( N = [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) -> j = k ) )
585582, 584sylib 208 . . . . . . . . . . . . . . . . 17 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) -> A. j e. ( 0 ... N ) A. k e. ( 0 ... N ) ( ( N = [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) -> j = k ) )
586 nfcsb1v 3549 . . . . . . . . . . . . . . . . . . . . 21 |- F/_ j [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C
587586nfeq2 2780 . . . . . . . . . . . . . . . . . . . 20 |- F/ j N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C
588 nfs1v 2437 . . . . . . . . . . . . . . . . . . . 20 |- F/ j [ k / j ] N = [_ ( 1st ` x ) / s ]_ C
589587, 588nfan 1828 . . . . . . . . . . . . . . . . . . 19 |- F/ j ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C )
590 nfv 1843 . . . . . . . . . . . . . . . . . . 19 |- F/ j ( 2nd ` x ) = k
591589, 590nfim 1825 . . . . . . . . . . . . . . . . . 18 |- F/ j ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = k )
592 nfv 1843 . . . . . . . . . . . . . . . . . 18 |- F/ k ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) )
593 csbeq1a 3542 . . . . . . . . . . . . . . . . . . . . 21 |- ( j = ( 2nd ` x ) -> [_ ( 1st ` x ) / s ]_ C = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C )
594593eqeq2d 2632 . . . . . . . . . . . . . . . . . . . 20 |- ( j = ( 2nd ` x ) -> ( N = [_ ( 1st ` x ) / s ]_ C <-> N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) )
595594anbi1d 741 . . . . . . . . . . . . . . . . . . 19 |- ( j = ( 2nd ` x ) -> ( ( N = [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) <-> ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) ) )
596 eqeq1 2626 . . . . . . . . . . . . . . . . . . 19 |- ( j = ( 2nd ` x ) -> ( j = k <-> ( 2nd ` x ) = k ) )
597595, 596imbi12d 334 . . . . . . . . . . . . . . . . . 18 |- ( j = ( 2nd ` x ) -> ( ( ( N = [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) -> j = k ) <-> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = k ) ) )
598 sbsbc 3439 . . . . . . . . . . . . . . . . . . . . . 22 |- ( [ k / j ] N = [_ ( 1st ` x ) / s ]_ C <-> [. k / j ]. N = [_ ( 1st ` x ) / s ]_ C )
599 vex 3203 . . . . . . . . . . . . . . . . . . . . . . 23 |- k e. _V
600 sbceq2g 3990 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( k e. _V -> ( [. k / j ]. N = [_ ( 1st ` x ) / s ]_ C <-> N = [_ k / j ]_ [_ ( 1st ` x ) / s ]_ C ) )
601599, 600ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 |- ( [. k / j ]. N = [_ ( 1st ` x ) / s ]_ C <-> N = [_ k / j ]_ [_ ( 1st ` x ) / s ]_ C )
602598, 601bitri 264 . . . . . . . . . . . . . . . . . . . . 21 |- ( [ k / j ] N = [_ ( 1st ` x ) / s ]_ C <-> N = [_ k / j ]_ [_ ( 1st ` x ) / s ]_ C )
603 csbeq1 3536 . . . . . . . . . . . . . . . . . . . . . 22 |- ( k = ( 2nd ` y ) -> [_ k / j ]_ [_ ( 1st ` x ) / s ]_ C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C )
604603eqeq2d 2632 . . . . . . . . . . . . . . . . . . . . 21 |- ( k = ( 2nd ` y ) -> ( N = [_ k / j ]_ [_ ( 1st ` x ) / s ]_ C <-> N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) )
605602, 604syl5bb 272 . . . . . . . . . . . . . . . . . . . 20 |- ( k = ( 2nd ` y ) -> ( [ k / j ] N = [_ ( 1st ` x ) / s ]_ C <-> N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) )
606605anbi2d 740 . . . . . . . . . . . . . . . . . . 19 |- ( k = ( 2nd ` y ) -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) <-> ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) ) )
607 eqeq2 2633 . . . . . . . . . . . . . . . . . . 19 |- ( k = ( 2nd ` y ) -> ( ( 2nd ` x ) = k <-> ( 2nd ` x ) = ( 2nd ` y ) ) )
608606, 607imbi12d 334 . . . . . . . . . . . . . . . . . 18 |- ( k = ( 2nd ` y ) -> ( ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = k ) <-> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) ) ) )
609591, 592, 597, 608rspc2 3320 . . . . . . . . . . . . . . . . 17 |- ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) -> ( A. j e. ( 0 ... N ) A. k e. ( 0 ... N ) ( ( N = [_ ( 1st ` x ) / s ]_ C /\ [ k / j ] N = [_ ( 1st ` x ) / s ]_ C ) -> j = k ) -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) ) ) )
610585, 609syl5com 31 . . . . . . . . . . . . . . . 16 |- ( ( ph /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C ) -> ( ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) ) ) )
611610impr 649 . . . . . . . . . . . . . . 15 |- ( ( ph /\ ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) ) ) -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) ) )
612 csbeq1 3536 . . . . . . . . . . . . . . . . . . 19 |- ( ( 1st ` x ) = ( 1st ` y ) -> [_ ( 1st ` x ) / s ]_ C = [_ ( 1st ` y ) / s ]_ C )
613612csbeq2dv 3992 . . . . . . . . . . . . . . . . . 18 |- ( ( 1st ` x ) = ( 1st ` y ) -> [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C )
614613eqeq2d 2632 . . . . . . . . . . . . . . . . 17 |- ( ( 1st ` x ) = ( 1st ` y ) -> ( N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C <-> N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) )
615614anbi2d 740 . . . . . . . . . . . . . . . 16 |- ( ( 1st ` x ) = ( 1st ` y ) -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) <-> ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) )
616615imbi1d 331 . . . . . . . . . . . . . . 15 |- ( ( 1st ` x ) = ( 1st ` y ) -> ( ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` x ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) ) <-> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) ) ) )
617611, 616syl5ibcom 235 . . . . . . . . . . . . . 14 |- ( ( ph /\ ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) -> ( 2nd ` x ) = ( 2nd ` y ) ) ) )
618617com23 86 . . . . . . . . . . . . 13 |- ( ( ph /\ ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) ) ) -> ( ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) -> ( ( 1st ` x ) = ( 1st ` y ) -> ( 2nd ` x ) = ( 2nd ` y ) ) ) )
619618impr 649 . . . . . . . . . . . 12 |- ( ( ph /\ ( ( A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 2nd ` x ) e. ( 0 ... N ) /\ ( 2nd ` y ) e. ( 0 ... N ) ) ) /\ ( N = [_ ( 2nd ` x ) / j ]_ [_ ( 1st ` x ) / s ]_ C /\ N = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / s ]_ C ) ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> ( 2nd ` x ) = ( 2nd ` y ) ) )
620573, 619sylan2 491 . . . . . . . . . . 11 |- ( ( ph /\ ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> ( 2nd ` x ) = ( 2nd ` y ) ) )
621 elrabi 3359 . . . . . . . . . . . . 13 |- ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -> x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
622 elrabi 3359 . . . . . . . . . . . . 13 |- ( y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -> y e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
623 xpopth 7207 . . . . . . . . . . . . . . 15 |- ( ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) <-> x = y ) )
624623biimpd 219 . . . . . . . . . . . . . 14 |- ( ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) -> x = y ) )
625624expd 452 . . . . . . . . . . . . 13 |- ( ( x e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> ( ( 2nd ` x ) = ( 2nd ` y ) -> x = y ) ) )
626621, 622, 625syl2an 494 . . . . . . . . . . . 12 |- ( ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) -> ( ( 1st ` x ) = ( 1st ` y ) -> ( ( 2nd ` x ) = ( 2nd ` y ) -> x = y ) ) )
627626adantl 482 . . . . . . . . . . 11 |- ( ( ph /\ ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> ( ( 2nd ` x ) = ( 2nd ` y ) -> x = y ) ) )
628620, 627mpdd 43 . . . . . . . . . 10 |- ( ( ph /\ ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> x = y ) )
629541, 628sylbid 230 . . . . . . . . 9 |- ( ( ph /\ ( x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } /\ y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) -> ( ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` y ) -> x = y ) )
630629ralrimivva 2971 . . . . . . . 8 |- ( ph -> A. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } A. y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` y ) -> x = y ) )
631 dff13 6512 . . . . . . . 8 |- ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -1-1-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } <-> ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } --> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } /\ A. x e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } A. y e. { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ( ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ` y ) -> x = y ) ) )
632537, 630, 631sylanbrc 698 . . . . . . 7 |- ( ph -> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -1-1-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
633 df-f1o 5895 . . . . . . 7 |- ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -1-1-onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } <-> ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -1-1-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } /\ ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) )
634632, 535, 633sylanbrc 698 . . . . . 6 |- ( ph -> ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -1-1-onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
635 rabfi 8185 . . . . . . . . 9 |- ( ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin -> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } e. Fin )
636138, 635ax-mp 5 . . . . . . . 8 |- { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } e. Fin
637636elexi 3213 . . . . . . 7 |- { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } e. _V
638637f1oen 7976 . . . . . 6 |- ( ( 1st |` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) : { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } -1-1-onto-> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } -> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ~~ { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
639634, 638syl 17 . . . . 5 |- ( ph -> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ~~ { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
640 rabfi 8185 . . . . . . 7 |- ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin -> { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } e. Fin )
641136, 640ax-mp 5 . . . . . 6 |- { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } e. Fin
642 hashen 13135 . . . . . 6 |- ( ( { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } e. Fin /\ { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } e. Fin ) -> ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = ( # ` { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) <-> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ~~ { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) )
643636, 641, 642mp2an 708 . . . . 5 |- ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = ( # ` { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) <-> { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ~~ { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } )
644639, 643sylibr 224 . . . 4 |- ( ph -> ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) = ( # ` { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) )
645644oveq2d 6666 . . 3 |- ( ph -> ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( N = [_ ( 2nd ` t ) / j ]_ [_ ( 1st ` t ) / s ]_ C /\ A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = [_ ( 1st ` t ) / s ]_ C ) } ) ) = ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) ) )
646202, 425, 6453eqtr3d 2664 . 2 |- ( ph -> sum_ x e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ( # ` ( { x } X. { y e. ( 0 ... N ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { y } ) i = [_ x / s ]_ C /\ A. j e. ( 0 ... N ) N =/= [_ x / s ]_ C ) } ) ) = ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) ) )
647135, 646breqtrd 4679 1 |- ( ph -> 2 || ( ( # ` { t e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | A. i e. ( 0 ... N ) E. j e. ( 0 ... N ) i = C } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   {cab 2608   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   E!wreu 2914   E*wrmo 2915   {crab 2916   _Vcvv 3200   [.wsbc 3435   [_csb 3533   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   iota_crio 6610  (class class class)co 6650   oFcof 6895   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117   sum_csu 14416   || cdvds 14983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984
This theorem is referenced by:  poimirlem28  33437
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