Theorem bnj1006 31029

Description: Technical lemma for bnj69 31078. 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
bnj1006.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1006.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1006.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1006.4	|- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
bnj1006.5	|- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
bnj1006.6	|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj1006.7	|- ( ph' <-> [. p / n ]. ph )
bnj1006.8	|- ( ps' <-> [. p / n ]. ps )
bnj1006.9	|- ( ch' <-> [. p / n ]. ch )
bnj1006.10	|- ( ph" <-> [. G / f ]. ph' )
bnj1006.11	|- ( ps" <-> [. G / f ]. ps' )
bnj1006.12	|- ( ch" <-> [. G / f ]. ch' )
bnj1006.13	|- D = ( om \ { (/) } )
bnj1006.15	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj1006.16	|- G = ( f u. { <. n , C >. } )
bnj1006.28	|- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. om /\ suc i e. p ) )

Assertion

Ref	Expression
bnj1006	|- ( ( th /\ ch /\ ta /\ et ) -> pred ( y , A , R ) C_ ( G ` suc i ) )

Distinct variable groups:   A, f, i, m, n, y   D, f, n   i, G   R, f, i, m, n, y   f, X, n   f, p, i, n

Allowed substitution hints:   ph( y, z, f, i, m, n, p)   ps( y, z, f, i, m, n, p)   ch( y, z, f, i, m, n, p)   th( y, z, f, i, m, n, p)   ta( y, z, f, i, m, n, p)   et( y, z, f, i, m, n, p)   A( z, p)   C( y, z, f, i, m, n, p)   D( y, z, i, m, p)   R( z, p)   G( y, z, f, m, n, p)   X( y, z, i, m, p)   ph'( y, z, f, i, m, n, p)   ps'( y, z, f, i, m, n, p)   ch'( y, z, f, i, m, n, p)   ph"( y, z, f, i, m, n, p)   ps"( y, z, f, i, m, n, p)   ch"( y, z, f, i, m, n, p)

Proof of Theorem bnj1006

Step	Hyp	Ref	Expression
1		bnj1006.6	. . . . 5 |- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
2	1	simprbi 480	. . . 4 |- ( et -> y e. ( f ` i ) )
3	2	bnj708 30826	. . 3 |- ( ( th /\ ch /\ ta /\ et ) -> y e. ( f ` i ) )
4		bnj1006.4	. . . . . . . 8 |- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
5		bnj253 30770	. . . . . . . . 9 |- ( ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) <-> ( ( R FrSe A /\ X e. A ) /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
6	5	simp1bi 1076	. . . . . . . 8 |- ( ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) -> ( R FrSe A /\ X e. A ) )
7	4, 6	sylbi 207	. . . . . . 7 |- ( th -> ( R FrSe A /\ X e. A ) )
8	7	bnj705 30823	. . . . . 6 |- ( ( th /\ ch /\ ta /\ et ) -> ( R FrSe A /\ X e. A ) )
9		bnj643 30819	. . . . . . 7 |- ( ( th /\ ch /\ ta /\ et ) -> ch )
10		bnj1006.5	. . . . . . . . 9 |- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
11		3simpc 1060	. . . . . . . . 9 |- ( ( m e. om /\ n = suc m /\ p = suc n ) -> ( n = suc m /\ p = suc n ) )
12	10, 11	sylbi 207	. . . . . . . 8 |- ( ta -> ( n = suc m /\ p = suc n ) )
13	12	bnj707 30825	. . . . . . 7 |- ( ( th /\ ch /\ ta /\ et ) -> ( n = suc m /\ p = suc n ) )
14		3anass 1042	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) <-> ( ch /\ ( n = suc m /\ p = suc n ) ) )
15	9, 13, 14	sylanbrc 698	. . . . . 6 |- ( ( th /\ ch /\ ta /\ et ) -> ( ch /\ n = suc m /\ p = suc n ) )
16		bnj1006.1	. . . . . . 7 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
17		bnj1006.2	. . . . . . 7 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
18		bnj1006.3	. . . . . . 7 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
19		bnj1006.13	. . . . . . 7 |- D = ( om \ { (/) } )
20		bnj1006.15	. . . . . . 7 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
21		biid 251	. . . . . . 7 |- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
22		biid 251	. . . . . . 7 |- ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ p = suc n /\ m e. n ) )
23	16, 17, 18, 19, 20, 21, 22	bnj969 31016	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
24	8, 15, 23	syl2anc 693	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> C e. _V )
25	18	bnj1235 30875	. . . . . 6 |- ( ch -> f Fn n )
26	25	bnj706 30824	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> f Fn n )
27	10	simp3bi 1078	. . . . . 6 |- ( ta -> p = suc n )
28	27	bnj707 30825	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> p = suc n )
29	1	simplbi 476	. . . . . 6 |- ( et -> i e. n )
30	29	bnj708 30826	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> i e. n )
31	24, 26, 28, 30	bnj951 30846	. . . 4 |- ( ( th /\ ch /\ ta /\ et ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) )
32		bnj1006.16	. . . . 5 |- G = ( f u. { <. n , C >. } )
33	32	bnj945 30844	. . . 4 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) -> ( G ` i ) = ( f ` i ) )
34	31, 33	syl 17	. . 3 |- ( ( th /\ ch /\ ta /\ et ) -> ( G ` i ) = ( f ` i ) )
35	3, 34	eleqtrrd 2704	. 2 |- ( ( th /\ ch /\ ta /\ et ) -> y e. ( G ` i ) )
36		bnj1006.28	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. om /\ suc i e. p ) )
37	36	anim1i 592	. . . 4 |- ( ( ( th /\ ch /\ ta /\ et ) /\ y e. ( G ` i ) ) -> ( ( ch" /\ i e. om /\ suc i e. p ) /\ y e. ( G ` i ) ) )
38		df-bnj17 30753	. . . 4 |- ( ( ch" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) <-> ( ( ch" /\ i e. om /\ suc i e. p ) /\ y e. ( G ` i ) ) )
39	37, 38	sylibr 224	. . 3 |- ( ( ( th /\ ch /\ ta /\ et ) /\ y e. ( G ` i ) ) -> ( ch" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) )
40		bnj1006.7	. . . 4 |- ( ph' <-> [. p / n ]. ph )
41		bnj1006.8	. . . 4 |- ( ps' <-> [. p / n ]. ps )
42		bnj1006.9	. . . 4 |- ( ch' <-> [. p / n ]. ch )
43		bnj1006.10	. . . 4 |- ( ph" <-> [. G / f ]. ph' )
44		bnj1006.11	. . . 4 |- ( ps" <-> [. G / f ]. ps' )
45		bnj1006.12	. . . 4 |- ( ch" <-> [. G / f ]. ch' )
46	16, 17, 18, 40, 41, 42, 43, 44, 45, 20, 32	bnj999 31027	. . 3 |- ( ( ch" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) -> pred ( y , A , R ) C_ ( G ` suc i ) )
47	39, 46	syl 17	. 2 |- ( ( ( th /\ ch /\ ta /\ et ) /\ y e. ( G ` i ) ) -> pred ( y , A , R ) C_ ( G ` suc i ) )
48	35, 47	mpdan 702	1 |- ( ( th /\ ch /\ ta /\ et ) -> pred ( y , A , R ) C_ ( G ` suc i ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-reu 2919  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj1020  31033