Theorem bnj1373 33793

Description: Technical lemma for bnj60 33825. 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
bnj1373.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1373.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1373.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1373.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1373.5	|- ( ta' <-> [. y / x ]. ta )

Assertion

Ref	Expression
bnj1373	|- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )

Distinct variable groups:   x, A   B, f   x, R   f, d, x   x, y

Allowed substitution hints:   ta( x, y, f, d)   A( y, f, d)   B( x, y, d)   C( x, y, f, d)   R( y, f, d)   G( x, y, f, d)   Y( x, y, f, d)   ta'( x, y, f, d)

Proof of Theorem bnj1373

Step	Hyp	Ref	Expression
1		bnj1373.5	. 2 |- ( ta' <-> [. y / x ]. ta )
2		vex 3096	. . 3 |- y e. _V
3		bnj1373.3	. . . . . . 7 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1373.1	. . . . . . . 8 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
5	4	bnj1309 33785	. . . . . . 7 |- ( f e. B -> A. x f e. B )
6	3, 5	bnj1307 33786	. . . . . 6 |- ( f e. C -> A. x f e. C )
7	6	bnj1351 33592	. . . . 5 |- ( ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) -> A. x ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
8	7	nfi 1608	. . . 4 |- F/ x ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) )
9		bnj1373.4	. . . . 5 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
10		sneq 4020	. . . . . . . 8 |- ( x = y -> { x } = { y } )
11		bnj1318 33788	. . . . . . . 8 |- ( x = y -> trCl ( x , A , R ) = trCl ( y , A , R ) )
12	10, 11	uneq12d 3641	. . . . . . 7 |- ( x = y -> ( { x } u. trCl ( x , A , R ) ) = ( { y } u. trCl ( y , A , R ) ) )
13	12	eqeq2d 2455	. . . . . 6 |- ( x = y -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom f = ( { y } u. trCl ( y , A , R ) ) ) )
14	13	anbi2d 703	. . . . 5 |- ( x = y -> ( ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
15	9, 14	syl5bb 257	. . . 4 |- ( x = y -> ( ta <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
16	8, 15	sbciegf 3343	. . 3 |- ( y e. _V -> ( [. y / x ]. ta <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
17	2, 16	ax-mp 5	. 2 |- ( [. y / x ]. ta <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
18	1, 17	bitri 249	1 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1381   e. wcel 1802   {cab 2426   A.wral 2791   E.wrex 2792   _Vcvv 3093   [.wsbc 3311   u. cun 3456   C_ wss 3458   {csn 4010   <.cop 4016   dom cdm 4985   |` cres 4987   Fn wfn 5569   ` cfv 5574   predc-bnj14 33448   trClc-bnj18 33454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-iun 4313  df-br 4434  df-bnj14 33449  df-bnj18 33455

This theorem is referenced by:  bnj1374  33794  bnj1384  33795  bnj1398  33797  bnj1450  33813  bnj1489  33819