Theorem fnwe2val 37619

Description: Lemma for fnwe2 37623. Substitute variables. (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 ) ) }

Assertion

Ref	Expression
fnwe2val	|- ( a T b <-> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) )

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

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

Proof of Theorem fnwe2val

Step	Hyp	Ref	Expression
1		vex 3203	. 2 |- a e. _V
2		vex 3203	. 2 |- b e. _V
3		fveq2 6191	. . . 4 |- ( x = a -> ( F ` x ) = ( F ` a ) )
4		fveq2 6191	. . . 4 |- ( y = b -> ( F ` y ) = ( F ` b ) )
5	3, 4	breqan12d 4669	. . 3 |- ( ( x = a /\ y = b ) -> ( ( F ` x ) R ( F ` y ) <-> ( F ` a ) R ( F ` b ) ) )
6	3, 4	eqeqan12d 2638	. . . 4 |- ( ( x = a /\ y = b ) -> ( ( F ` x ) = ( F ` y ) <-> ( F ` a ) = ( F ` b ) ) )
7		simpl 473	. . . . 5 |- ( ( x = a /\ y = b ) -> x = a )
8		fvex 6201	. . . . . . . 8 |- ( F ` x ) e. _V
9		fnwe2.su	. . . . . . . 8 |- ( z = ( F ` x ) -> S = U )
10	8, 9	csbie 3559	. . . . . . 7 |- [_ ( F ` x ) / z ]_ S = U
11	3	csbeq1d 3540	. . . . . . 7 |- ( x = a -> [_ ( F ` x ) / z ]_ S = [_ ( F ` a ) / z ]_ S )
12	10, 11	syl5eqr 2670	. . . . . 6 |- ( x = a -> U = [_ ( F ` a ) / z ]_ S )
13	12	adantr 481	. . . . 5 |- ( ( x = a /\ y = b ) -> U = [_ ( F ` a ) / z ]_ S )
14		simpr 477	. . . . 5 |- ( ( x = a /\ y = b ) -> y = b )
15	7, 13, 14	breq123d 4667	. . . 4 |- ( ( x = a /\ y = b ) -> ( x U y <-> a [_ ( F ` a ) / z ]_ S b ) )
16	6, 15	anbi12d 747	. . 3 |- ( ( x = a /\ y = b ) -> ( ( ( F ` x ) = ( F ` y ) /\ x U y ) <-> ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) )
17	5, 16	orbi12d 746	. 2 |- ( ( x = a /\ y = b ) -> ( ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) <-> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) ) )
18		fnwe2.t	. 2 |- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) }
19	1, 2, 17, 18	braba 4992	1 |- ( a T b <-> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   [_csb 3533   class class class wbr 4653   {copab 4712   ` 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-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  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-opab 4713  df-iota 5851  df-fv 5896

This theorem is referenced by:  fnwe2lem2  37621  fnwe2lem3  37622