Theorem bnj1373 29911

Description: Technical lemma for bnj60 29943. 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 3034	. . 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 29903	. . . . . . 7 |- ( f e. B -> A. x f e. B )
6	3, 5	bnj1307 29904	. . . . . 6 |- ( f e. C -> A. x f e. C )
7	6	bnj1351 29710	. . . . 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 1682	. . . 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 3969	. . . . . . . 8 |- ( x = y -> { x } = { y } )
11		bnj1318 29906	. . . . . . . 8 |- ( x = y -> trCl ( x , A , R ) = trCl ( y , A , R ) )
12	10, 11	uneq12d 3580	. . . . . . 7 |- ( x = y -> ( { x } u. trCl ( x , A , R ) ) = ( { y } u. trCl ( y , A , R ) ) )
13	12	eqeq2d 2481	. . . . . 6 |- ( x = y -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom f = ( { y } u. trCl ( y , A , R ) ) ) )
14	13	anbi2d 718	. . . . 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 265	. . . 4 |- ( x = y -> ( ta <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
16	8, 15	sbciegf 3287	. . 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 257	1 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   _Vcvv 3031   [.wsbc 3255   u. cun 3388   C_ wss 3390   {csn 3959   <.cop 3965   dom cdm 4839   |` cres 4841   Fn wfn 5584   ` cfv 5589   predc-bnj14 29565   trClc-bnj18 29571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4271  df-br 4396  df-bnj14 29566  df-bnj18 29572

This theorem is referenced by:  bnj1374  29912  bnj1384  29913  bnj1398  29915  bnj1450  29931  bnj1489  29937