Theorem bnj1052 29832

Description: Technical lemma for bnj69 29867. 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
bnj1052.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1052.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1052.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1052.4	|- ( th <-> ( R FrSe A /\ X e. A ) )
bnj1052.5	|- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
bnj1052.6	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
bnj1052.7	|- D = ( om \ { (/) } )
bnj1052.8	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1052.9	|- ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
bnj1052.10	|- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
bnj1052.37	|- ( ( th /\ ta /\ ch /\ ze ) -> ( _E Fr n /\ A. i e. n ( rh -> et ) ) )

Assertion

Ref	Expression
bnj1052	|- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )

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

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

Proof of Theorem bnj1052

Step	Hyp	Ref	Expression
1		bnj1052.1	. 2 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		bnj1052.2	. 2 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj1052.3	. 2 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4		bnj1052.4	. 2 |- ( th <-> ( R FrSe A /\ X e. A ) )
5		bnj1052.5	. 2 |- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
6		bnj1052.6	. 2 |- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
7		bnj1052.7	. 2 |- D = ( om \ { (/) } )
8		bnj1052.8	. 2 |- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9		19.23vv 1829	. . . . 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 1701	. . . 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 1828	. . . 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 257	. . 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		bnj1052.37	. . . . 5 |- ( ( th /\ ta /\ ch /\ ze ) -> ( _E Fr n /\ A. i e. n ( rh -> et ) ) )
14		vex 3059	. . . . . . . . 9 |- n e. _V
15		bnj1052.10	. . . . . . . . 9 |- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
16	14, 15	bnj110 29717	. . . . . . . 8 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i e. n et )
17		bnj1052.9	. . . . . . . . 9 |- ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
18	6, 17	bnj1049 29831	. . . . . . . 8 |- ( A. i e. n et <-> A. i et )
19	16, 18	sylib 201	. . . . . . 7 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i et )
20	19	19.21bi 1957	. . . . . 6 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> et )
21	20, 17	sylib 201	. . . . 5 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
22	13, 21	mpcom 37	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
23	22	gen2 1680	. . 3 |- A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
24	12, 23	mpgbi 1682	. 2 |- ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )
25	1, 2, 3, 4, 5, 6, 7, 8, 24	bnj1034 29827	1 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   A.wal 1452   = wceq 1454   E.wex 1673   e. wcel 1897   {cab 2447   A.wral 2748   E.wrex 2749   _Vcvv 3056   [.wsbc 3278   \ cdif 3412   C_ wss 3415   (/)c0 3742   {csn 3979   U_ciun 4291   class class class wbr 4415   _E cep 4761   Fr wfr 4808   suc csuc 5443   Fn wfn 5595   ` cfv 5600   omcom 6718   /\ w-bnj17 29539   predc-bnj14 29541   FrSe w-bnj15 29545   trClc-bnj18 29547   TrFow-bnj19 29549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-iun 4293  df-br 4416  df-fr 4811  df-fn 5603  df-bnj17 29540  df-bnj18 29548

This theorem is referenced by:  bnj1053  29833