Theorem bnj1018 31032

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
bnj1018.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1018.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1018.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1018.4	|- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
bnj1018.5	|- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
bnj1018.7	|- ( ph' <-> [. p / n ]. ph )
bnj1018.8	|- ( ps' <-> [. p / n ]. ps )
bnj1018.9	|- ( ch' <-> [. p / n ]. ch )
bnj1018.10	|- ( ph" <-> [. G / f ]. ph' )
bnj1018.11	|- ( ps" <-> [. G / f ]. ps' )
bnj1018.12	|- ( ch" <-> [. G / f ]. ch' )
bnj1018.13	|- D = ( om \ { (/) } )
bnj1018.14	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1018.15	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj1018.16	|- G = ( f u. { <. n , C >. } )
bnj1018.26	|- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
bnj1018.29	|- ( ( th /\ ch /\ ta /\ et ) -> ch" )
bnj1018.30	|- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. om /\ suc i e. p ) )

Assertion

Ref	Expression
bnj1018	|- ( ( th /\ ch /\ et /\ E. p ta ) -> ( G ` suc i ) C_ trCl ( X , A , R ) )

Distinct variable groups:   A, f, i, m, n, y   D, f, i, n   i, G, p   R, f, i, m, n, y   f, X, i, n, y   ch, p   et, p   f, p, n   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, p)   B( y, z, f, i, m, n, p)   C( y, z, f, i, m, n, p)   D( y, z, m, p)   R( z, p)   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 bnj1018

Step	Hyp	Ref	Expression
1		df-bnj17 30753	. . 3 |- ( ( th /\ ch /\ et /\ E. p ta ) <-> ( ( th /\ ch /\ et ) /\ E. p ta ) )
2		bnj258 30774	. . . . . . . 8 |- ( ( th /\ ch /\ ta /\ et ) <-> ( ( th /\ ch /\ et ) /\ ta ) )
3		bnj1018.29	. . . . . . . 8 |- ( ( th /\ ch /\ ta /\ et ) -> ch" )
4	2, 3	sylbir 225	. . . . . . 7 |- ( ( ( th /\ ch /\ et ) /\ ta ) -> ch" )
5	4	ex 450	. . . . . 6 |- ( ( th /\ ch /\ et ) -> ( ta -> ch" ) )
6	5	eximdv 1846	. . . . 5 |- ( ( th /\ ch /\ et ) -> ( E. p ta -> E. p ch" ) )
7		bnj1018.3	. . . . . 6 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
8		bnj1018.9	. . . . . 6 |- ( ch' <-> [. p / n ]. ch )
9		bnj1018.12	. . . . . 6 |- ( ch" <-> [. G / f ]. ch' )
10		bnj1018.14	. . . . . 6 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
11		bnj1018.16	. . . . . 6 |- G = ( f u. { <. n , C >. } )
12	7, 8, 9, 10, 11	bnj985 31023	. . . . 5 |- ( G e. B <-> E. p ch" )
13	6, 12	syl6ibr 242	. . . 4 |- ( ( th /\ ch /\ et ) -> ( E. p ta -> G e. B ) )
14	13	imp 445	. . 3 |- ( ( ( th /\ ch /\ et ) /\ E. p ta ) -> G e. B )
15	1, 14	sylbi 207	. 2 |- ( ( th /\ ch /\ et /\ E. p ta ) -> G e. B )
16		bnj1019 30850	. . 3 |- ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )
17		bnj1018.30	. . . . . 6 |- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. om /\ suc i e. p ) )
18	17	simp3d 1075	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> suc i e. p )
19		bnj1018.26	. . . . . . 7 |- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
20	19	bnj1235 30875	. . . . . 6 |- ( ch" -> G Fn p )
21		fndm 5990	. . . . . 6 |- ( G Fn p -> dom G = p )
22	3, 20, 21	3syl 18	. . . . 5 |- ( ( th /\ ch /\ ta /\ et ) -> dom G = p )
23	18, 22	eleqtrrd 2704	. . . 4 |- ( ( th /\ ch /\ ta /\ et ) -> suc i e. dom G )
24	23	exlimiv 1858	. . 3 |- ( E. p ( th /\ ch /\ ta /\ et ) -> suc i e. dom G )
25	16, 24	sylbir 225	. 2 |- ( ( th /\ ch /\ et /\ E. p ta ) -> suc i e. dom G )
26		bnj1018.1	. . 3 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
27		bnj1018.2	. . 3 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
28		bnj1018.13	. . 3 |- D = ( om \ { (/) } )
29	11	bnj918 30836	. . 3 |- G e. _V
30		vex 3203	. . . 4 |- i e. _V
31	30	sucex 7011	. . 3 |- suc i e. _V
32	26, 27, 28, 10, 29, 31	bnj1015 31031	. 2 |- ( ( G e. B /\ suc i e. dom G ) -> ( G ` suc i ) C_ trCl ( X , A , R ) )
33	15, 25, 32	syl2anc 693	1 |- ( ( th /\ ch /\ et /\ E. p ta ) -> ( G ` suc i ) 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   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   dom cdm 5114   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-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-dm 5124  df-suc 5729  df-iota 5851  df-fn 5891  df-fv 5896  df-bnj17 30753  df-bnj18 30761

This theorem is referenced by:  bnj1020  31033