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Theorem catcfuccl 16759
Description: The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcfuccl.c |- C = (CatCat ` U )
catcfuccl.b |- B = ( Base ` C )
catcfuccl.o |- Q = ( X FuncCat Y )
catcfuccl.u |- ( ph -> U e. WUni )
catcfuccl.1 |- ( ph -> om e. U )
catcfuccl.x |- ( ph -> X e. B )
catcfuccl.y |- ( ph -> Y e. B )
Assertion
Ref Expression
catcfuccl |- ( ph -> Q e. B )
Proof of Theorem catcfuccl
Dummy variables a b f g h v x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcfuccl.o . . . . 5 |- Q = ( X FuncCat Y )
2 eqid 2622 . . . . 5 |- ( X Func Y ) = ( X Func Y )
3 eqid 2622 . . . . 5 |- ( X Nat Y ) = ( X Nat Y )
4 eqid 2622 . . . . 5 |- ( Base ` X ) = ( Base ` X )
5 eqid 2622 . . . . 5 |- (comp ` Y ) = (comp ` Y )
6 inss2 3834 . . . . . 6 |- ( U i^i Cat ) C_ Cat
7 catcfuccl.x . . . . . . 7 |- ( ph -> X e. B )
8 catcfuccl.c . . . . . . . 8 |- C = (CatCat ` U )
9 catcfuccl.b . . . . . . . 8 |- B = ( Base ` C )
10 catcfuccl.u . . . . . . . 8 |- ( ph -> U e. WUni )
118, 9, 10catcbas 16747 . . . . . . 7 |- ( ph -> B = ( U i^i Cat ) )
127, 11eleqtrd 2703 . . . . . 6 |- ( ph -> X e. ( U i^i Cat ) )
136, 12sseldi 3601 . . . . 5 |- ( ph -> X e. Cat )
14 catcfuccl.y . . . . . . 7 |- ( ph -> Y e. B )
1514, 11eleqtrd 2703 . . . . . 6 |- ( ph -> Y e. ( U i^i Cat ) )
166, 15sseldi 3601 . . . . 5 |- ( ph -> Y e. Cat )
17 eqidd 2623 . . . . 5 |- ( ph -> ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
181, 2, 3, 4, 5, 13, 16, 17fucval 16618 . . . 4 |- ( ph -> Q = { <. ( Base ` ndx ) , ( X Func Y ) >. , <. ( Hom ` ndx ) , ( X Nat Y ) >. , <. (comp ` ndx ) , ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
19 df-base 15863 . . . . . . 7 |- Base = Slot 1
20 catcfuccl.1 . . . . . . . 8 |- ( ph -> om e. U )
2110, 20wunndx 15878 . . . . . . 7 |- ( ph -> ndx e. U )
2219, 10, 21wunstr 15881 . . . . . 6 |- ( ph -> ( Base ` ndx ) e. U )
23 inss1 3833 . . . . . . . 8 |- ( U i^i Cat ) C_ U
2423, 12sseldi 3601 . . . . . . 7 |- ( ph -> X e. U )
2523, 15sseldi 3601 . . . . . . 7 |- ( ph -> Y e. U )
2610, 24, 25wunfunc 16559 . . . . . 6 |- ( ph -> ( X Func Y ) e. U )
2710, 22, 26wunop 9544 . . . . 5 |- ( ph -> <. ( Base ` ndx ) , ( X Func Y ) >. e. U )
28 df-hom 15966 . . . . . . 7 |- Hom = Slot ; 1 4
2928, 10, 21wunstr 15881 . . . . . 6 |- ( ph -> ( Hom ` ndx ) e. U )
3010, 24, 25wunnat 16616 . . . . . 6 |- ( ph -> ( X Nat Y ) e. U )
3110, 29, 30wunop 9544 . . . . 5 |- ( ph -> <. ( Hom ` ndx ) , ( X Nat Y ) >. e. U )
32 df-cco 15967 . . . . . . 7 |- comp = Slot ; 1 5
3332, 10, 21wunstr 15881 . . . . . 6 |- ( ph -> (comp ` ndx ) e. U )
3410, 26, 26wunxp 9546 . . . . . . . 8 |- ( ph -> ( ( X Func Y ) X. ( X Func Y ) ) e. U )
3510, 34, 26wunxp 9546 . . . . . . 7 |- ( ph -> ( ( ( X Func Y ) X. ( X Func Y ) ) X. ( X Func Y ) ) e. U )
3632, 10, 25wunstr 15881 . . . . . . . . . . . . . 14 |- ( ph -> (comp ` Y ) e. U )
3710, 36wunrn 9551 . . . . . . . . . . . . 13 |- ( ph -> ran (comp ` Y ) e. U )
3810, 37wununi 9528 . . . . . . . . . . . 12 |- ( ph -> U. ran (comp ` Y ) e. U )
3910, 38wunrn 9551 . . . . . . . . . . 11 |- ( ph -> ran U. ran (comp ` Y ) e. U )
4010, 39wununi 9528 . . . . . . . . . 10 |- ( ph -> U. ran U. ran (comp ` Y ) e. U )
4110, 40wunpw 9529 . . . . . . . . 9 |- ( ph -> ~P U. ran U. ran (comp ` Y ) e. U )
4219, 10, 24wunstr 15881 . . . . . . . . 9 |- ( ph -> ( Base ` X ) e. U )
4310, 41, 42wunmap 9548 . . . . . . . 8 |- ( ph -> ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) e. U )
4410, 30wunrn 9551 . . . . . . . . . 10 |- ( ph -> ran ( X Nat Y ) e. U )
4510, 44wununi 9528 . . . . . . . . 9 |- ( ph -> U. ran ( X Nat Y ) e. U )
4610, 45, 45wunxp 9546 . . . . . . . 8 |- ( ph -> ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) e. U )
4710, 43, 46wunpm 9547 . . . . . . 7 |- ( ph -> ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) e. U )
48 fvex 6201 . . . . . . . . . . 11 |- ( 1st ` v ) e. _V
49 fvex 6201 . . . . . . . . . . . . . 14 |- ( 2nd ` v ) e. _V
50 ovex 6678 . . . . . . . . . . . . . . . . 17 |- ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) e. _V
51 ovex 6678 . . . . . . . . . . . . . . . . . . . 20 |- ( X Nat Y ) e. _V
5251rnex 7100 . . . . . . . . . . . . . . . . . . 19 |- ran ( X Nat Y ) e. _V
5352uniex 6953 . . . . . . . . . . . . . . . . . 18 |- U. ran ( X Nat Y ) e. _V
5453, 53xpex 6962 . . . . . . . . . . . . . . . . 17 |- ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) e. _V
55 eqid 2622 . . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) = ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) )
56 ovssunirn 6681 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) C_ U. ran ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) )
57 ovssunirn 6681 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) C_ U. ran (comp ` Y )
58 rnss 5354 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) C_ U. ran (comp ` Y ) -> ran ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) C_ ran U. ran (comp ` Y ) )
59 uniss 4458 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ran ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) C_ ran U. ran (comp ` Y ) -> U. ran ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) C_ U. ran U. ran (comp ` Y ) )
6057, 58, 59mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24 |- U. ran ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) C_ U. ran U. ran (comp ` Y )
6156, 60sstri 3612 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) C_ U. ran U. ran (comp ` Y )
62 ovex 6678 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) e. _V
6362elpw 4164 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) e. ~P U. ran U. ran (comp ` Y ) <-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) C_ U. ran U. ran (comp ` Y ) )
6461, 63mpbir 221 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) e. ~P U. ran U. ran (comp ` Y )
6564a1i 11 . . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( Base ` X ) -> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) e. ~P U. ran U. ran (comp ` Y ) )
6655, 65fmpti 6383 . . . . . . . . . . . . . . . . . . . 20 |- ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) : ( Base ` X ) --> ~P U. ran U. ran (comp ` Y )
67 fvex 6201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (comp ` Y ) e. _V
6867rnex 7100 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ran (comp ` Y ) e. _V
6968uniex 6953 . . . . . . . . . . . . . . . . . . . . . . . 24 |- U. ran (comp ` Y ) e. _V
7069rnex 7100 . . . . . . . . . . . . . . . . . . . . . . 23 |- ran U. ran (comp ` Y ) e. _V
7170uniex 6953 . . . . . . . . . . . . . . . . . . . . . 22 |- U. ran U. ran (comp ` Y ) e. _V
7271pwex 4848 . . . . . . . . . . . . . . . . . . . . 21 |- ~P U. ran U. ran (comp ` Y ) e. _V
73 fvex 6201 . . . . . . . . . . . . . . . . . . . . 21 |- ( Base ` X ) e. _V
7472, 73elmap 7886 . . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) e. ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) <-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) : ( Base ` X ) --> ~P U. ran U. ran (comp ` Y ) )
7566, 74mpbir 221 . . . . . . . . . . . . . . . . . . 19 |- ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) e. ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) )
7675rgen2w 2925 . . . . . . . . . . . . . . . . . 18 |- A. b e. ( g ( X Nat Y ) h ) A. a e. ( f ( X Nat Y ) g ) ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) e. ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) )
77 eqid 2622 . . . . . . . . . . . . . . . . . . 19 |- ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) )
7877fmpt2 7237 . . . . . . . . . . . . . . . . . 18 |- ( A. b e. ( g ( X Nat Y ) h ) A. a e. ( f ( X Nat Y ) g ) ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) e. ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) <-> ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) : ( ( g ( X Nat Y ) h ) X. ( f ( X Nat Y ) g ) ) --> ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) )
7976, 78mpbi 220 . . . . . . . . . . . . . . . . 17 |- ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) : ( ( g ( X Nat Y ) h ) X. ( f ( X Nat Y ) g ) ) --> ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) )
80 ovssunirn 6681 . . . . . . . . . . . . . . . . . 18 |- ( g ( X Nat Y ) h ) C_ U. ran ( X Nat Y )
81 ovssunirn 6681 . . . . . . . . . . . . . . . . . 18 |- ( f ( X Nat Y ) g ) C_ U. ran ( X Nat Y )
82 xpss12 5225 . . . . . . . . . . . . . . . . . 18 |- ( ( ( g ( X Nat Y ) h ) C_ U. ran ( X Nat Y ) /\ ( f ( X Nat Y ) g ) C_ U. ran ( X Nat Y ) ) -> ( ( g ( X Nat Y ) h ) X. ( f ( X Nat Y ) g ) ) C_ ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) )
8380, 81, 82mp2an 708 . . . . . . . . . . . . . . . . 17 |- ( ( g ( X Nat Y ) h ) X. ( f ( X Nat Y ) g ) ) C_ ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) )
84 elpm2r 7875 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) e. _V /\ ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) e. _V ) /\ ( ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) : ( ( g ( X Nat Y ) h ) X. ( f ( X Nat Y ) g ) ) --> ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) /\ ( ( g ( X Nat Y ) h ) X. ( f ( X Nat Y ) g ) ) C_ ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) ) -> ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) )
8550, 54, 79, 83, 84mp4an 709 . . . . . . . . . . . . . . . 16 |- ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) )
8685sbcth 3450 . . . . . . . . . . . . . . 15 |- ( ( 2nd ` v ) e. _V -> [. ( 2nd ` v ) / g ]. ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) )
87 sbcel1g 3987 . . . . . . . . . . . . . . 15 |- ( ( 2nd ` v ) e. _V -> ( [. ( 2nd ` v ) / g ]. ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) <-> [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) ) )
8886, 87mpbid 222 . . . . . . . . . . . . . 14 |- ( ( 2nd ` v ) e. _V -> [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) )
8949, 88ax-mp 5 . . . . . . . . . . . . 13 |- [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) )
9089sbcth 3450 . . . . . . . . . . . 12 |- ( ( 1st ` v ) e. _V -> [. ( 1st ` v ) / f ]. [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) )
91 sbcel1g 3987 . . . . . . . . . . . 12 |- ( ( 1st ` v ) e. _V -> ( [. ( 1st ` v ) / f ]. [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) <-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) ) )
9290, 91mpbid 222 . . . . . . . . . . 11 |- ( ( 1st ` v ) e. _V -> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) )
9348, 92ax-mp 5 . . . . . . . . . 10 |- [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) )
9493rgen2w 2925 . . . . . . . . 9 |- A. v e. ( ( X Func Y ) X. ( X Func Y ) ) A. h e. ( X Func Y ) [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) )
95 eqid 2622 . . . . . . . . . 10 |- ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
9695fmpt2 7237 . . . . . . . . 9 |- ( A. v e. ( ( X Func Y ) X. ( X Func Y ) ) A. h e. ( X Func Y ) [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) e. ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) <-> ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) : ( ( ( X Func Y ) X. ( X Func Y ) ) X. ( X Func Y ) ) --> ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) )
9794, 96mpbi 220 . . . . . . . 8 |- ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) : ( ( ( X Func Y ) X. ( X Func Y ) ) X. ( X Func Y ) ) --> ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) )
9897a1i 11 . . . . . . 7 |- ( ph -> ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) : ( ( ( X Func Y ) X. ( X Func Y ) ) X. ( X Func Y ) ) --> ( ( ~P U. ran U. ran (comp ` Y ) ^m ( Base ` X ) ) ^pm ( U. ran ( X Nat Y ) X. U. ran ( X Nat Y ) ) ) )
9910, 35, 47, 98wunf 9549 . . . . . 6 |- ( ph -> ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) e. U )
10010, 33, 99wunop 9544 . . . . 5 |- ( ph -> <. (comp ` ndx ) , ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. e. U )
10110, 27, 31, 100wuntp 9533 . . . 4 |- ( ph -> { <. ( Base ` ndx ) , ( X Func Y ) >. , <. ( Hom ` ndx ) , ( X Nat Y ) >. , <. (comp ` ndx ) , ( v e. ( ( X Func Y ) X. ( X Func Y ) ) , h e. ( X Func Y ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( X Nat Y ) h ) , a e. ( f ( X Nat Y ) g ) |-> ( x e. ( Base ` X ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` Y ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } e. U )
10218, 101eqeltrd 2701 . . 3 |- ( ph -> Q e. U )
1031, 13, 16fuccat 16630 . . 3 |- ( ph -> Q e. Cat )
104102, 103elind 3798 . 2 |- ( ph -> Q e. ( U i^i Cat ) )
105104, 11eleqtrrd 2704 1 |- ( ph -> Q e. B )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   [_csb 3533   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {ctp 4181   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   ^pm cpm 7858  WUnicwun 9522   1c1 9937   4c4 11072   5c5 11073  ;cdc 11493   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602  CatCatccatc 16744
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
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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wun 9524  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877  df-nr 9878  df-c 9942  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-func 16518  df-nat 16603  df-fuc 16604  df-catc 16745
This theorem is referenced by: (None)
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