Theorem bnj580 30983

Description: Technical lemma for bnj579 30984. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj580.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj580.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj580.3	|- ( ch <-> ( f Fn n /\ ph /\ ps ) )
bnj580.4	|- ( ph' <-> [. g / f ]. ph )
bnj580.5	|- ( ps' <-> [. g / f ]. ps )
bnj580.6	|- ( ch' <-> [. g / f ]. ch )
bnj580.7	|- D = ( om \ { (/) } )
bnj580.8	|- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
bnj580.9	|- ( ta <-> A. k e. n ( k _E j -> [. k / j ]. th ) )

Assertion

Ref	Expression
bnj580	|- ( n e. D -> E* f ch )

Distinct variable groups:   A, f, i, k   D, f, g, j, k   R, f, i, k   ch, g, j, k   j, ch', k   f, n   g, i, n, k   x, f   y, f, g, i, k   j, n   th, k

Allowed substitution hints:   ph( x, y, f, g, i, j, k, n)   ps( x, y, f, g, i, j, k, n)   ch( x, y, f, i, n)   th( x, y, f, g, i, j, n)   ta( x, y, f, g, i, j, k, n)   A( x, y, g, j, n)   D( x, y, i, n)   R( x, y, g, j, n)   ph'( x, y, f, g, i, j, k, n)   ps'( x, y, f, g, i, j, k, n)   ch'( x, y, f, g, i, n)

Proof of Theorem bnj580

Step	Hyp	Ref	Expression
1		bnj580.3	. . . . . . 7 |- ( ch <-> ( f Fn n /\ ph /\ ps ) )
2	1	simp1bi 1076	. . . . . 6 |- ( ch -> f Fn n )
3		bnj580.4	. . . . . . . 8 |- ( ph' <-> [. g / f ]. ph )
4		bnj580.5	. . . . . . . 8 |- ( ps' <-> [. g / f ]. ps )
5		bnj580.6	. . . . . . . 8 |- ( ch' <-> [. g / f ]. ch )
6	1, 3, 4, 5	bnj581 30978	. . . . . . 7 |- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )
7	6	simp1bi 1076	. . . . . 6 |- ( ch' -> g Fn n )
8	2, 7	bnj240 30765	. . . . 5 |- ( ( n e. D /\ ch /\ ch' ) -> ( f Fn n /\ g Fn n ) )
9		bnj580.1	. . . . . . . . . . . . 13 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
10		bnj580.2	. . . . . . . . . . . . 13 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
11		bnj580.7	. . . . . . . . . . . . 13 |- D = ( om \ { (/) } )
12	3, 9	bnj154 30948	. . . . . . . . . . . . 13 |- ( ph' <-> ( g ` (/) ) = pred ( x , A , R ) )
13		vex 3203	. . . . . . . . . . . . . 14 |- g e. _V
14	10, 4, 13	bnj540 30962	. . . . . . . . . . . . 13 |- ( ps' <-> A. i e. om ( suc i e. n -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) )
15		bnj580.8	. . . . . . . . . . . . 13 |- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
16	15	bnj591 30981	. . . . . . . . . . . . 13 |- ( [. k / j ]. th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) )
17		bnj580.9	. . . . . . . . . . . . 13 |- ( ta <-> A. k e. n ( k _E j -> [. k / j ]. th ) )
18	9, 10, 1, 11, 12, 14, 6, 15, 16, 17	bnj594 30982	. . . . . . . . . . . 12 |- ( ( j e. n /\ ta ) -> th )
19	18	ex 450	. . . . . . . . . . 11 |- ( j e. n -> ( ta -> th ) )
20	19	rgen 2922	. . . . . . . . . 10 |- A. j e. n ( ta -> th )
21		vex 3203	. . . . . . . . . . 11 |- n e. _V
22	21, 17	bnj110 30928	. . . . . . . . . 10 |- ( ( _E Fr n /\ A. j e. n ( ta -> th ) ) -> A. j e. n th )
23	20, 22	mpan2 707	. . . . . . . . 9 |- ( _E Fr n -> A. j e. n th )
24	15	ralbii 2980	. . . . . . . . 9 |- ( A. j e. n th <-> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
25	23, 24	sylib 208	. . . . . . . 8 |- ( _E Fr n -> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
26	25	r19.21be 2933	. . . . . . 7 |- A. j e. n ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
27	11	bnj923 30838	. . . . . . . . . . . . 13 |- ( n e. D -> n e. om )
28		nnord 7073	. . . . . . . . . . . . 13 |- ( n e. om -> Ord n )
29		ordfr 5738	. . . . . . . . . . . . 13 |- ( Ord n -> _E Fr n )
30	27, 28, 29	3syl 18	. . . . . . . . . . . 12 |- ( n e. D -> _E Fr n )
31	30	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( n e. D /\ ch /\ ch' ) -> _E Fr n )
32	31	pm4.71ri 665	. . . . . . . . . 10 |- ( ( n e. D /\ ch /\ ch' ) <-> ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) )
33	32	imbi1i 339	. . . . . . . . 9 |- ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) )
34		impexp 462	. . . . . . . . 9 |- ( ( ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) <-> ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) )
35	33, 34	bitri 264	. . . . . . . 8 |- ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) )
36	35	ralbii 2980	. . . . . . 7 |- ( A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> A. j e. n ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) )
37	26, 36	mpbir 221	. . . . . 6 |- A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) )
38		r19.21v 2960	. . . . . 6 |- ( A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ch /\ ch' ) -> A. j e. n ( f ` j ) = ( g ` j ) ) )
39	37, 38	mpbi 220	. . . . 5 |- ( ( n e. D /\ ch /\ ch' ) -> A. j e. n ( f ` j ) = ( g ` j ) )
40		eqfnfv 6311	. . . . . 6 |- ( ( f Fn n /\ g Fn n ) -> ( f = g <-> A. j e. n ( f ` j ) = ( g ` j ) ) )
41	40	biimprd 238	. . . . 5 |- ( ( f Fn n /\ g Fn n ) -> ( A. j e. n ( f ` j ) = ( g ` j ) -> f = g ) )
42	8, 39, 41	sylc 65	. . . 4 |- ( ( n e. D /\ ch /\ ch' ) -> f = g )
43	42	3expib 1268	. . 3 |- ( n e. D -> ( ( ch /\ ch' ) -> f = g ) )
44	43	alrimivv 1856	. 2 |- ( n e. D -> A. f A. g ( ( ch /\ ch' ) -> f = g ) )
45		sbsbc 3439	. . . . . 6 |- ( [ g / f ] ch <-> [. g / f ]. ch )
46	45	anbi2i 730	. . . . 5 |- ( ( ch /\ [ g / f ] ch ) <-> ( ch /\ [. g / f ]. ch ) )
47	46	imbi1i 339	. . . 4 |- ( ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
48	47	2albii 1748	. . 3 |- ( A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
49		nfv 1843	. . . 4 |- F/ g ch
50	49	mo3 2507	. . 3 |- ( E* f ch <-> A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) )
51	5	anbi2i 730	. . . . 5 |- ( ( ch /\ ch' ) <-> ( ch /\ [. g / f ]. ch ) )
52	51	imbi1i 339	. . . 4 |- ( ( ( ch /\ ch' ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
53	52	2albii 1748	. . 3 |- ( A. f A. g ( ( ch /\ ch' ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
54	48, 50, 53	3bitr4i 292	. 2 |- ( E* f ch <-> A. f A. g ( ( ch /\ ch' ) -> f = g ) )
55	44, 54	sylibr 224	1 |- ( n e. D -> E* f ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   [wsb 1880   e. wcel 1990   E*wmo 2471   A.wral 2912   [.wsbc 3435   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   _E cep 5028   Fr wfr 5070   Ord word 5722   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   predc-bnj14 30754

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  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-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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj579  30984