Theorem bnj907 31035

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
bnj907.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj907.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj907.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj907.4	|- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
bnj907.5	|- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
bnj907.6	|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj907.7	|- ( ph' <-> [. p / n ]. ph )
bnj907.8	|- ( ps' <-> [. p / n ]. ps )
bnj907.9	|- ( ch' <-> [. p / n ]. ch )
bnj907.10	|- ( ph" <-> [. G / f ]. ph' )
bnj907.11	|- ( ps" <-> [. G / f ]. ps' )
bnj907.12	|- ( ch" <-> [. G / f ]. ch' )
bnj907.13	|- D = ( om \ { (/) } )
bnj907.14	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj907.15	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj907.16	|- G = ( f u. { <. n , C >. } )

Assertion

Ref	Expression
bnj907	|- ( ( R FrSe A /\ X e. A ) -> TrFo ( trCl ( X , A , R ) , A , R ) )

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

Allowed substitution hints:   ph( y, z, f, m, n, p)   ps( y, z, f, i, m, n, p)   ch( y, z, f, i, n)   th( y, z)   ta( y, z, f, i, m, n, p)   et( y, z, f, i, n)   B( y, z, f, i, m, n, p)   C( y, z, f, i, m, n, p)   D( y, z, m, p)   G( y, z, f, m, n)   X( 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 bnj907

Step	Hyp	Ref	Expression
1		bnj907.4	. 2 |- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
2		bnj907.1	. . . . . . . . 9 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
3		bnj907.2	. . . . . . . . 9 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
4		bnj907.3	. . . . . . . . 9 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
5		bnj907.5	. . . . . . . . 9 |- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
6		bnj907.6	. . . . . . . . 9 |- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
7		bnj907.13	. . . . . . . . 9 |- D = ( om \ { (/) } )
8		bnj907.14	. . . . . . . . 9 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9	2, 3, 4, 1, 5, 6, 7, 8	bnj1021 31034	. . . . . . . 8 |- E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) )
10		bnj907.7	. . . . . . . . . . . 12 |- ( ph' <-> [. p / n ]. ph )
11		bnj907.8	. . . . . . . . . . . 12 |- ( ps' <-> [. p / n ]. ps )
12		bnj907.9	. . . . . . . . . . . 12 |- ( ch' <-> [. p / n ]. ch )
13		bnj907.10	. . . . . . . . . . . 12 |- ( ph" <-> [. G / f ]. ph' )
14		bnj907.11	. . . . . . . . . . . 12 |- ( ps" <-> [. G / f ]. ps' )
15		bnj907.12	. . . . . . . . . . . 12 |- ( ch" <-> [. G / f ]. ch' )
16		bnj907.15	. . . . . . . . . . . 12 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
17		bnj907.16	. . . . . . . . . . . 12 |- G = ( f u. { <. n , C >. } )
18		vex 3203	. . . . . . . . . . . . . 14 |- p e. _V
19	4, 10, 11, 12, 18	bnj919 30837	. . . . . . . . . . . . 13 |- ( ch' <-> ( p e. D /\ f Fn p /\ ph' /\ ps' ) )
20	17	bnj918 30836	. . . . . . . . . . . . 13 |- G e. _V
21	19, 13, 14, 15, 20	bnj976 30848	. . . . . . . . . . . 12 |- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
22	2, 3, 4, 1, 5, 6, 10, 11, 12, 13, 14, 15, 7, 8, 16, 17, 21	bnj1020 31033	. . . . . . . . . . 11 |- ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) )
23	22	ax-gen 1722	. . . . . . . . . 10 |- A. m ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) )
24		19.29r 1802	. . . . . . . . . . 11 |- ( ( E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ A. m ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) ) -> E. m ( ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) ) )
25		pm3.33 609	. . . . . . . . . . 11 |- ( ( ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) ) -> ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
26	24, 25	bnj593 30815	. . . . . . . . . 10 |- ( ( E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ A. m ( ( th /\ ch /\ et /\ E. p ta ) -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) ) -> E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
27	23, 26	mpan2 707	. . . . . . . . 9 |- ( E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) -> E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
28	27	2eximi 1763	. . . . . . . 8 |- ( E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) -> E. n E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
29	9, 28	bnj101 30789	. . . . . . 7 |- E. f E. n E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) )
30		19.9v 1896	. . . . . . 7 |- ( E. f E. n E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) <-> E. n E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
31	29, 30	mpbi 220	. . . . . 6 |- E. n E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) )
32		19.9v 1896	. . . . . 6 |- ( E. n E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) <-> E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
33	31, 32	mpbi 220	. . . . 5 |- E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) )
34		19.9v 1896	. . . . 5 |- ( E. i E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) <-> E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
35	33, 34	mpbi 220	. . . 4 |- E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) )
36		19.9v 1896	. . . 4 |- ( E. m ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) <-> ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) ) )
37	35, 36	mpbi 220	. . 3 |- ( th -> pred ( y , A , R ) C_ trCl ( X , A , R ) )
38	1	bnj1254 30880	. . 3 |- ( th -> z e. pred ( y , A , R ) )
39	37, 38	sseldd 3604	. 2 |- ( th -> z e. trCl ( X , A , R ) )
40	1, 39	bnj978 31019	1 |- ( ( R FrSe A /\ X e. A ) -> TrFo ( trCl ( X , A , R ) , A , R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = 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   TrFow-bnj19 30762

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  df-bnj19 30763

This theorem is referenced by:  bnj1029  31036