Theorem bnj1052 34451

Description: Technical lemma for bnj69 34486. 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 1766	. . . . 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 1645	. . . 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 1765	. . . 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 249	. . 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 3109	. . . . . . . . 9 |- n e. _V
15		bnj1052.10	. . . . . . . . 9 |- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
16	14, 15	bnj110 34336	. . . . . . . 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 34450	. . . . . . . 8 |- ( A. i e. n et <-> A. i et )
19	16, 18	sylib 196	. . . . . . 7 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i et )
20	19	19.21bi 1874	. . . . . 6 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> et )
21	20, 17	sylib 196	. . . . 5 |- ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
22	13, 21	mpcom 36	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
23	22	gen2 1624	. . 3 |- A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
24	12, 23	mpgbi 1626	. 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 34446	1 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   A.wal 1396   = wceq 1398   E.wex 1617   e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805   _Vcvv 3106   [.wsbc 3324   \ cdif 3458   C_ wss 3461   (/)c0 3783   {csn 4016   U_ciun 4315   class class class wbr 4439   _E cep 4778   Fr wfr 4824   suc csuc 4869   Fn wfn 5565   ` cfv 5570   omcom 6673   /\ w-bnj17 34158   predc-bnj14 34160   FrSe w-bnj15 34164   trClc-bnj18 34166   TrFow-bnj19 34168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-iun 4317  df-br 4440  df-fr 4827  df-fn 5573  df-bnj17 34159  df-bnj18 34167

This theorem is referenced by:  bnj1053  34452