Theorem fpwwecbv 9466

Description: Lemma for fpwwe 9468. (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 473	. . . . . 6 |- ( ( x = a /\ r = s ) -> x = a )
3	2	sseq1d 3632	. . . . 5 |- ( ( x = a /\ r = s ) -> ( x C_ A <-> a C_ A ) )
4		simpr 477	. . . . . 6 |- ( ( x = a /\ r = s ) -> r = s )
5	2	sqxpeqd 5141	. . . . . 6 |- ( ( x = a /\ r = s ) -> ( x X. x ) = ( a X. a ) )
6	4, 5	sseq12d 3634	. . . . 5 |- ( ( x = a /\ r = s ) -> ( r C_ ( x X. x ) <-> s C_ ( a X. a ) ) )
7	3, 6	anbi12d 747	. . . 4 |- ( ( x = a /\ r = s ) -> ( ( x C_ A /\ r C_ ( x X. x ) ) <-> ( a C_ A /\ s C_ ( a X. a ) ) ) )
8		weeq2 5103	. . . . . 6 |- ( x = a -> ( r We x <-> r We a ) )
9		weeq1 5102	. . . . . 6 |- ( r = s -> ( r We a <-> s We a ) )
10	8, 9	sylan9bb 736	. . . . 5 |- ( ( x = a /\ r = s ) -> ( r We x <-> s We a ) )
11		sneq 4187	. . . . . . . . . 10 |- ( y = z -> { y } = { z } )
12	11	imaeq2d 5466	. . . . . . . . 9 |- ( y = z -> ( `' r " { y } ) = ( `' r " { z } ) )
13	12	fveq2d 6195	. . . . . . . 8 |- ( y = z -> ( F ` ( `' r " { y } ) ) = ( F ` ( `' r " { z } ) ) )
14		id 22	. . . . . . . 8 |- ( y = z -> y = z )
15	13, 14	eqeq12d 2637	. . . . . . 7 |- ( y = z -> ( ( F ` ( `' r " { y } ) ) = y <-> ( F ` ( `' r " { z } ) ) = z ) )
16	15	cbvralv 3171	. . . . . 6 |- ( A. y e. x ( F ` ( `' r " { y } ) ) = y <-> A. z e. x ( F ` ( `' r " { z } ) ) = z )
17	4	cnveqd 5298	. . . . . . . . . 10 |- ( ( x = a /\ r = s ) -> `' r = `' s )
18	17	imaeq1d 5465	. . . . . . . . 9 |- ( ( x = a /\ r = s ) -> ( `' r " { z } ) = ( `' s " { z } ) )
19	18	fveq2d 6195	. . . . . . . 8 |- ( ( x = a /\ r = s ) -> ( F ` ( `' r " { z } ) ) = ( F ` ( `' s " { z } ) ) )
20	19	eqeq1d 2624	. . . . . . 7 |- ( ( x = a /\ r = s ) -> ( ( F ` ( `' r " { z } ) ) = z <-> ( F ` ( `' s " { z } ) ) = z ) )
21	2, 20	raleqbidv 3152	. . . . . 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 272	. . . . 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 747	. . . 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 747	. . 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 4722	. 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 2644	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 384   = wceq 1483   A.wral 2912   C_ wss 3574   {csn 4177   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   "cima 5117   ` 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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896

This theorem is referenced by:  canthnum  9471  canthp1  9476