Theorem fpwwecbv 9087

Description: Lemma for fpwwe 9089. (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 464	. . . . . 6 |- ( ( x = a /\ r = s ) -> x = a )
3	2	sseq1d 3445	. . . . 5 |- ( ( x = a /\ r = s ) -> ( x C_ A <-> a C_ A ) )
4		simpr 468	. . . . . 6 |- ( ( x = a /\ r = s ) -> r = s )
5	2	sqxpeqd 4865	. . . . . 6 |- ( ( x = a /\ r = s ) -> ( x X. x ) = ( a X. a ) )
6	4, 5	sseq12d 3447	. . . . 5 |- ( ( x = a /\ r = s ) -> ( r C_ ( x X. x ) <-> s C_ ( a X. a ) ) )
7	3, 6	anbi12d 725	. . . 4 |- ( ( x = a /\ r = s ) -> ( ( x C_ A /\ r C_ ( x X. x ) ) <-> ( a C_ A /\ s C_ ( a X. a ) ) ) )
8		weeq2 4828	. . . . . 6 |- ( x = a -> ( r We x <-> r We a ) )
9		weeq1 4827	. . . . . 6 |- ( r = s -> ( r We a <-> s We a ) )
10	8, 9	sylan9bb 714	. . . . 5 |- ( ( x = a /\ r = s ) -> ( r We x <-> s We a ) )
11		sneq 3969	. . . . . . . . . 10 |- ( y = z -> { y } = { z } )
12	11	imaeq2d 5174	. . . . . . . . 9 |- ( y = z -> ( `' r " { y } ) = ( `' r " { z } ) )
13	12	fveq2d 5883	. . . . . . . 8 |- ( y = z -> ( F ` ( `' r " { y } ) ) = ( F ` ( `' r " { z } ) ) )
14		id 22	. . . . . . . 8 |- ( y = z -> y = z )
15	13, 14	eqeq12d 2486	. . . . . . 7 |- ( y = z -> ( ( F ` ( `' r " { y } ) ) = y <-> ( F ` ( `' r " { z } ) ) = z ) )
16	15	cbvralv 3005	. . . . . 6 |- ( A. y e. x ( F ` ( `' r " { y } ) ) = y <-> A. z e. x ( F ` ( `' r " { z } ) ) = z )
17	4	cnveqd 5015	. . . . . . . . . 10 |- ( ( x = a /\ r = s ) -> `' r = `' s )
18	17	imaeq1d 5173	. . . . . . . . 9 |- ( ( x = a /\ r = s ) -> ( `' r " { z } ) = ( `' s " { z } ) )
19	18	fveq2d 5883	. . . . . . . 8 |- ( ( x = a /\ r = s ) -> ( F ` ( `' r " { z } ) ) = ( F ` ( `' s " { z } ) ) )
20	19	eqeq1d 2473	. . . . . . 7 |- ( ( x = a /\ r = s ) -> ( ( F ` ( `' r " { z } ) ) = z <-> ( F ` ( `' s " { z } ) ) = z ) )
21	2, 20	raleqbidv 2987	. . . . . 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 265	. . . . 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 725	. . . 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 725	. . 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 4465	. 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 2493	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 376   = wceq 1452   A.wral 2756   C_ wss 3390   {csn 3959   {copab 4453   We wwe 4797   X. cxp 4837   `'ccnv 4838   "cima 4842   ` cfv 5589

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-3or 1008  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-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-uni 4191  df-br 4396  df-opab 4455  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fv 5597

This theorem is referenced by:  canthnum  9092  canthp1  9097