Theorem fnwe2lem1 37620

Description: Lemma for fnwe2 37623. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
fnwe2.su	|- ( z = ( F ` x ) -> S = U )
fnwe2.t	|- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) }
fnwe2.s	|- ( ( ph /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )

Assertion

Ref	Expression
fnwe2lem1	|- ( ( ph /\ a e. A ) -> [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } )

Distinct variable groups:   y, U, z, a   x, S, y, a   x, R, y, a   ph, x, y, z   x, A, y, z, a   x, F, y, z, a   T, a

Allowed substitution hints:   ph( a)   R( z)   S( z)   T( x, y, z)   U( x)

Proof of Theorem fnwe2lem1

Step	Hyp	Ref	Expression
1		fnwe2.s	. . . 4 |- ( ( ph /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )
2	1	ralrimiva 2966	. . 3 |- ( ph -> A. x e. A U We { y e. A | ( F ` y ) = ( F ` x ) } )
3		fveq2 6191	. . . . . . 7 |- ( a = x -> ( F ` a ) = ( F ` x ) )
4	3	csbeq1d 3540	. . . . . 6 |- ( a = x -> [_ ( F ` a ) / z ]_ S = [_ ( F ` x ) / z ]_ S )
5		fvex 6201	. . . . . . 7 |- ( F ` x ) e. _V
6		fnwe2.su	. . . . . . 7 |- ( z = ( F ` x ) -> S = U )
7	5, 6	csbie 3559	. . . . . 6 |- [_ ( F ` x ) / z ]_ S = U
8	4, 7	syl6eq 2672	. . . . 5 |- ( a = x -> [_ ( F ` a ) / z ]_ S = U )
9	3	eqeq2d 2632	. . . . . 6 |- ( a = x -> ( ( F ` y ) = ( F ` a ) <-> ( F ` y ) = ( F ` x ) ) )
10	9	rabbidv 3189	. . . . 5 |- ( a = x -> { y e. A | ( F ` y ) = ( F ` a ) } = { y e. A | ( F ` y ) = ( F ` x ) } )
11	8, 10	weeq12d 37610	. . . 4 |- ( a = x -> ( [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } <-> U We { y e. A | ( F ` y ) = ( F ` x ) } ) )
12	11	cbvralv 3171	. . 3 |- ( A. a e. A [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } <-> A. x e. A U We { y e. A | ( F ` y ) = ( F ` x ) } )
13	2, 12	sylibr 224	. 2 |- ( ph -> A. a e. A [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } )
14	13	r19.21bi 2932	1 |- ( ( ph /\ a e. A ) -> [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   [_csb 3533   class class class wbr 4653   {copab 4712   We wwe 5072   ` cfv 5888

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  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  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-csb 3534  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-uni 4437  df-br 4654  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-iota 5851  df-fv 5896

This theorem is referenced by:  fnwe2lem2  37621  fnwe2lem3  37622