Theorem bnj1373 31098

Description: Technical lemma for bnj60 31130. 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 3203	. . 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 31090	. . . . . . 7 |- ( f e. B -> A. x f e. B )
6	3, 5	bnj1307 31091	. . . . . 6 |- ( f e. C -> A. x f e. C )
7	6	bnj1351 30897	. . . . 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	nf5i 2024	. . . 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 4187	. . . . . . . 8 |- ( x = y -> { x } = { y } )
11		bnj1318 31093	. . . . . . . 8 |- ( x = y -> trCl ( x , A , R ) = trCl ( y , A , R ) )
12	10, 11	uneq12d 3768	. . . . . . 7 |- ( x = y -> ( { x } u. trCl ( x , A , R ) ) = ( { y } u. trCl ( y , A , R ) ) )
13	12	eqeq2d 2632	. . . . . 6 |- ( x = y -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom f = ( { y } u. trCl ( y , A , R ) ) ) )
14	13	anbi2d 740	. . . . 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 272	. . . 4 |- ( x = y -> ( ta <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
16	8, 15	sbciegf 3467	. . 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 264	1 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   u. cun 3572   C_ wss 3574   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   trClc-bnj18 30760

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-bnj14 30755  df-bnj18 30761

This theorem is referenced by:  bnj1374  31099  bnj1384  31100  bnj1398  31102  bnj1450  31118  bnj1489  31124