Theorem fpwwecbv 9011

Description: Lemma for fpwwe 9013. (Contributed by Mario Carneiro, 15-May-2015.)

Hypothesis

Ref	Expression
fpwwe.1	|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }

Assertion

Ref	Expression
fpwwecbv	|- W = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( F ` ( `' s " { z } ) ) = z ) ) }

Distinct variable groups:   r, a, s, x, A   y, a, z, F, r, s, x

Allowed substitution hints:   A( y, z)   W( x, y, z, s, r, a)

Proof of Theorem fpwwecbv

Step	Hyp	Ref	Expression
1		fpwwe.1	. 2 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }
2		simpl 455	. . . . . 6 |- ( ( x = a /\ r = s ) -> x = a )
3	2	sseq1d 3516	. . . . 5 |- ( ( x = a /\ r = s ) -> ( x C_ A <-> a C_ A ) )
4		simpr 459	. . . . . 6 |- ( ( x = a /\ r = s ) -> r = s )
5	2	sqxpeqd 5014	. . . . . 6 |- ( ( x = a /\ r = s ) -> ( x X. x ) = ( a X. a ) )
6	4, 5	sseq12d 3518	. . . . 5 |- ( ( x = a /\ r = s ) -> ( r C_ ( x X. x ) <-> s C_ ( a X. a ) ) )
7	3, 6	anbi12d 708	. . . 4 |- ( ( x = a /\ r = s ) -> ( ( x C_ A /\ r C_ ( x X. x ) ) <-> ( a C_ A /\ s C_ ( a X. a ) ) ) )
8		weeq2 4857	. . . . . 6 |- ( x = a -> ( r We x <-> r We a ) )
9		weeq1 4856	. . . . . 6 |- ( r = s -> ( r We a <-> s We a ) )
10	8, 9	sylan9bb 697	. . . . 5 |- ( ( x = a /\ r = s ) -> ( r We x <-> s We a ) )
11		sneq 4026	. . . . . . . . . 10 |- ( y = z -> { y } = { z } )
12	11	imaeq2d 5325	. . . . . . . . 9 |- ( y = z -> ( `' r " { y } ) = ( `' r " { z } ) )
13	12	fveq2d 5852	. . . . . . . 8 |- ( y = z -> ( F ` ( `' r " { y } ) ) = ( F ` ( `' r " { z } ) ) )
14		id 22	. . . . . . . 8 |- ( y = z -> y = z )
15	13, 14	eqeq12d 2476	. . . . . . 7 |- ( y = z -> ( ( F ` ( `' r " { y } ) ) = y <-> ( F ` ( `' r " { z } ) ) = z ) )
16	15	cbvralv 3081	. . . . . 6 |- ( A. y e. x ( F ` ( `' r " { y } ) ) = y <-> A. z e. x ( F ` ( `' r " { z } ) ) = z )
17	4	cnveqd 5167	. . . . . . . . . 10 |- ( ( x = a /\ r = s ) -> `' r = `' s )
18	17	imaeq1d 5324	. . . . . . . . 9 |- ( ( x = a /\ r = s ) -> ( `' r " { z } ) = ( `' s " { z } ) )
19	18	fveq2d 5852	. . . . . . . 8 |- ( ( x = a /\ r = s ) -> ( F ` ( `' r " { z } ) ) = ( F ` ( `' s " { z } ) ) )
20	19	eqeq1d 2456	. . . . . . 7 |- ( ( x = a /\ r = s ) -> ( ( F ` ( `' r " { z } ) ) = z <-> ( F ` ( `' s " { z } ) ) = z ) )
21	2, 20	raleqbidv 3065	. . . . . 6 |- ( ( x = a /\ r = s ) -> ( A. z e. x ( F ` ( `' r " { z } ) ) = z <-> A. z e. a ( F ` ( `' s " { z } ) ) = z ) )
22	16, 21	syl5bb 257	. . . . 5 |- ( ( x = a /\ r = s ) -> ( A. y e. x ( F ` ( `' r " { y } ) ) = y <-> A. z e. a ( F ` ( `' s " { z } ) ) = z ) )
23	10, 22	anbi12d 708	. . . 4 |- ( ( x = a /\ r = s ) -> ( ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) <-> ( s We a /\ A. z e. a ( F ` ( `' s " { z } ) ) = z ) ) )
24	7, 23	anbi12d 708	. . 3 |- ( ( x = a /\ r = s ) -> ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) <-> ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( F ` ( `' s " { z } ) ) = z ) ) ) )
25	24	cbvopabv 4508	. 2 |- { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) } = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( F ` ( `' s " { z } ) ) = z ) ) }
26	1, 25	eqtri 2483	1 |- W = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( F ` ( `' s " { z } ) ) = z ) ) }

Syntax hints:   /\ wa 367   = wceq 1398   A.wral 2804   C_ wss 3461   {csn 4016   {copab 4496   We wwe 4826   X. cxp 4986   `'ccnv 4987   "cima 4991   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578

This theorem is referenced by:  canthnum  9016  canthp1  9021