Theorem bnj1373 32026

Description: Technical lemma for bnj60 32058. 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 2980	. . 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 32018	. . . . . . 7 |- ( f e. B -> A. x f e. B )
6	3, 5	bnj1307 32019	. . . . . 6 |- ( f e. C -> A. x f e. C )
7	6	bnj1351 31825	. . . . 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 1596	. . . 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 3892	. . . . . . . 8 |- ( x = y -> { x } = { y } )
11		bnj1318 32021	. . . . . . . 8 |- ( x = y -> trCl ( x , A , R ) = trCl ( y , A , R ) )
12	10, 11	uneq12d 3516	. . . . . . 7 |- ( x = y -> ( { x } u. trCl ( x , A , R ) ) = ( { y } u. trCl ( y , A , R ) ) )
13	12	eqeq2d 2454	. . . . . 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 3223	. . 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 1369   e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721   _Vcvv 2977   [.wsbc 3191   u. cun 3331   C_ wss 3333   {csn 3882   <.cop 3888   dom cdm 4845   |` cres 4847   Fn wfn 5418   ` cfv 5423   predc-bnj14 31681   trClc-bnj18 31687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-iun 4178  df-br 4298  df-bnj14 31682  df-bnj18 31688

This theorem is referenced by:  bnj1374  32027  bnj1384  32028  bnj1398  32030  bnj1450  32046  bnj1489  32052