Theorem bnj1030 31055

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
bnj1030.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1030.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1030.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1030.4	|- ( th <-> ( R FrSe A /\ X e. A ) )
bnj1030.5	|- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
bnj1030.6	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
bnj1030.7	|- D = ( om \ { (/) } )
bnj1030.8	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1030.9	|- ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
bnj1030.10	|- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
bnj1030.11	|- ( ph' <-> [. j / i ]. ph )
bnj1030.12	|- ( ps' <-> [. j / i ]. ps )
bnj1030.13	|- ( ch' <-> [. j / i ]. ch )
bnj1030.14	|- ( th' <-> [. j / i ]. th )
bnj1030.15	|- ( ta' <-> [. j / i ]. ta )
bnj1030.16	|- ( ze' <-> [. j / i ]. ze )
bnj1030.17	|- ( et' <-> [. j / i ]. et )
bnj1030.18	|- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
bnj1030.19	|- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )

Assertion

Ref	Expression
bnj1030	|- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )

Distinct variable groups:   A, f, i, j, n, y   z, A, f, i, n   B, f, i, n, y   z, B   D, i, j   R, f, i, j, n, y   z, R   f, X, i, n, y   z, X   ch, j   et, j   ta, f, i, j, n   th, f, i, j, n   ph, i   ta, z   th, z

Allowed substitution hints:   ph( y, z, f, j, n)   ps( y, z, f, i, j, n)   ch( y, z, f, i, n)   th( y)   ta( y)   et( y, z, f, i, n)   ze( y, z, f, i, j, n)   si( y, z, f, i, j, n)   rh( y, z, f, i, j, n)   B( j)   D( y, z, f, n)   K( y, z, f, i, j, n)   X( j)   ph'( y, z, f, i, j, n)   ps'( y, z, f, i, j, n)   ch'( y, z, f, i, j, n)   th'( y, z, f, i, j, n)   ta'( y, z, f, i, j, n)   et'( y, z, f, i, j, n)   ze'( y, z, f, i, j, n)   ph0( y, z, f, i, j, n)

Proof of Theorem bnj1030

Step	Hyp	Ref	Expression
1		bnj1030.1	. 2 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		bnj1030.2	. 2 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj1030.3	. 2 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4		bnj1030.4	. 2 |- ( th <-> ( R FrSe A /\ X e. A ) )
5		bnj1030.5	. 2 |- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
6		bnj1030.6	. 2 |- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
7		bnj1030.7	. 2 |- D = ( om \ { (/) } )
8		bnj1030.8	. 2 |- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9		19.23vv 1903	. . . . 5 |- ( A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> ( E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
10	9	albii 1747	. . . 4 |- ( A. f A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> A. f ( E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
11		19.23v 1902	. . . 4 |- ( A. f ( E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
12	10, 11	bitri 264	. . 3 |- ( A. f A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
13		bnj1030.9	. . . . 5 |- ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
14	7	bnj1071 31045	. . . . . . . 8 |- ( n e. D -> _E Fr n )
15	3, 14	bnj769 30832	. . . . . . 7 |- ( ch -> _E Fr n )
16	15	bnj707 30825	. . . . . 6 |- ( ( th /\ ta /\ ch /\ ze ) -> _E Fr n )
17		bnj1030.10	. . . . . . 7 |- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
18		bnj1030.17	. . . . . . 7 |- ( et' <-> [. j / i ]. et )
19		bnj1030.18	. . . . . . 7 |- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
20		bnj1030.19	. . . . . . 7 |- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )
21	2, 8, 13, 18	bnj1123 31054	. . . . . . . . . 10 |- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )
22	2, 3, 5, 7, 19, 20, 21	bnj1118 31052	. . . . . . . . 9 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )
23	1, 3, 5	bnj1097 31049	. . . . . . . . 9 |- ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )
24	22, 23	bnj1109 30857	. . . . . . . 8 |- E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B )
25	24, 2, 3	bnj1093 31048	. . . . . . 7 |- ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )
26	13, 17, 18, 19, 20, 25	bnj1090 31047	. . . . . 6 |- ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n ( rh -> et ) )
27		vex 3203	. . . . . . 7 |- n e. _V
28	27, 17	bnj110 30928	. . . . . 6 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i e. n et )
29	16, 26, 28	syl2anc 693	. . . . 5 |- ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n et )
30	4, 5, 3, 6, 13, 29, 8	bnj1121 31053	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
31	30	gen2 1723	. . 3 |- A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
32	12, 31	mpgbi 1725	. 2 |- ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )
33	1, 2, 3, 4, 5, 6, 7, 8, 32	bnj1034 31038	1 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )

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   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   _E cep 5028   Fr wfr 5070   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   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-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-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-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1124  31056