Theorem bnj1052 31043

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
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 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		bnj1052.37	. . . . 5 |- ( ( th /\ ta /\ ch /\ ze ) -> ( _E Fr n /\ A. i e. n ( rh -> et ) ) )
14		vex 3203	. . . . . . . . 9 |- n e. _V
15		bnj1052.10	. . . . . . . . 9 |- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
16	14, 15	bnj110 30928	. . . . . . . 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 31042	. . . . . . . 8 |- ( A. i e. n et <-> A. i et )
19	16, 18	sylib 208	. . . . . . 7 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i et )
20	19	19.21bi 2059	. . . . . 6 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> et )
21	20, 17	sylib 208	. . . . 5 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
22	13, 21	mpcom 38	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
23	22	gen2 1723	. . 3 |- A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
24	12, 23	mpgbi 1725	. 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 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   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-br 4654  df-fr 5073  df-fn 5891  df-bnj17 30753  df-bnj18 30761

This theorem is referenced by:  bnj1053  31044