Theorem bnj1020 31033

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
bnj1020.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1020.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1020.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1020.4	|- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
bnj1020.5	|- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
bnj1020.6	|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj1020.7	|- ( ph' <-> [. p / n ]. ph )
bnj1020.8	|- ( ps' <-> [. p / n ]. ps )
bnj1020.9	|- ( ch' <-> [. p / n ]. ch )
bnj1020.10	|- ( ph" <-> [. G / f ]. ph' )
bnj1020.11	|- ( ps" <-> [. G / f ]. ps' )
bnj1020.12	|- ( ch" <-> [. G / f ]. ch' )
bnj1020.13	|- D = ( om \ { (/) } )
bnj1020.14	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1020.15	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj1020.16	|- G = ( f u. { <. n , C >. } )
bnj1020.26	|- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )

Assertion

Ref	Expression
bnj1020	|- ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) )

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

Allowed substitution hints:   ph( y, z, f, m, n, p)   ps( y, z, f, i, m, n, p)   ch( y, z, f, i, m, n)   th( y, z, f, i, m, n)   ta( y, z, f, i, m, n, p)   et( y, z, f, i, m, n)   A( z)   B( y, z, f, i, m, n, p)   C( y, z, f, i, m, n, p)   D( y, z, m, p)   R( z)   G( y, z, f, m, n)   X( z, 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 bnj1020

Step	Hyp	Ref	Expression
1		bnj1019 30850	. . 3 |- ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )
2		bnj1020.1	. . . . 5 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
3		bnj1020.2	. . . . 5 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
4		bnj1020.3	. . . . 5 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
5		bnj1020.4	. . . . 5 |- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
6		bnj1020.5	. . . . 5 |- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
7		bnj1020.6	. . . . 5 |- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
8		bnj1020.7	. . . . 5 |- ( ph' <-> [. p / n ]. ph )
9		bnj1020.8	. . . . 5 |- ( ps' <-> [. p / n ]. ps )
10		bnj1020.9	. . . . 5 |- ( ch' <-> [. p / n ]. ch )
11		bnj1020.10	. . . . 5 |- ( ph" <-> [. G / f ]. ph' )
12		bnj1020.11	. . . . 5 |- ( ps" <-> [. G / f ]. ps' )
13		bnj1020.12	. . . . 5 |- ( ch" <-> [. G / f ]. ch' )
14		bnj1020.13	. . . . 5 |- D = ( om \ { (/) } )
15		bnj1020.15	. . . . 5 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
16		bnj1020.16	. . . . 5 |- G = ( f u. { <. n , C >. } )
17		bnj1020.14	. . . . . . 7 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
18	2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16	bnj998 31026	. . . . . 6 |- ( ( th /\ ch /\ ta /\ et ) -> ch" )
19	4, 6, 7, 14, 18	bnj1001 31028	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. om /\ suc i e. p ) )
20	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19	bnj1006 31029	. . . 4 |- ( ( th /\ ch /\ ta /\ et ) -> pred ( y , A , R ) C_ ( G ` suc i ) )
21	20	exlimiv 1858	. . 3 |- ( E. p ( th /\ ch /\ ta /\ et ) -> pred ( y , A , R ) C_ ( G ` suc i ) )
22	1, 21	sylbir 225	. 2 |- ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ ( G ` suc i ) )
23		bnj1020.26	. . 3 |- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
24	2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16, 23, 18, 19	bnj1018 31032	. 2 |- ( ( th /\ ch /\ et /\ E. p ta ) -> ( G ` suc i ) C_ trCl ( X , A , R ) )
25	22, 24	sstrd 3613	1 |- ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   [.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  df-bnj18 30761

This theorem is referenced by:  bnj907  31035