Theorem fpwwe2lem2 9454

Description: Lemma for fpwwe2 9465. (Contributed by Mario Carneiro, 19-May-2015.)

Hypotheses

Ref	Expression
fpwwe2.1	|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }
fpwwe2.2	|- ( ph -> A e. _V )

Assertion

Ref	Expression
fpwwe2lem2	|- ( ph -> ( X W R <-> ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) )

Distinct variable groups:   y, u, r, x, F   X, r, u, x, y   ph, r, u, x, y   A, r, x   R, r, u, x, y   W, r, u, x, y

Allowed substitution hints:   A( y, u)

Proof of Theorem fpwwe2lem2

Step	Hyp	Ref	Expression
1		fpwwe2.1	. . . . 5 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }
2	1	relopabi 5245	. . . 4 |- Rel W
3	2	a1i 11	. . 3 |- ( ph -> Rel W )
4		brrelex12 5155	. . 3 |- ( ( Rel W /\ X W R ) -> ( X e. _V /\ R e. _V ) )
5	3, 4	sylan 488	. 2 |- ( ( ph /\ X W R ) -> ( X e. _V /\ R e. _V ) )
6		fpwwe2.2	. . . . 5 |- ( ph -> A e. _V )
7	6	adantr 481	. . . 4 |- ( ( ph /\ ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) -> A e. _V )
8		simprll 802	. . . 4 |- ( ( ph /\ ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) -> X C_ A )
9	7, 8	ssexd 4805	. . 3 |- ( ( ph /\ ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) -> X e. _V )
10		xpexg 6960	. . . . 5 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
11	9, 9, 10	syl2anc 693	. . . 4 |- ( ( ph /\ ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) -> ( X X. X ) e. _V )
12		simprlr 803	. . . 4 |- ( ( ph /\ ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) -> R C_ ( X X. X ) )
13	11, 12	ssexd 4805	. . 3 |- ( ( ph /\ ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) -> R e. _V )
14	9, 13	jca 554	. 2 |- ( ( ph /\ ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) -> ( X e. _V /\ R e. _V ) )
15		simpl 473	. . . . . 6 |- ( ( x = X /\ r = R ) -> x = X )
16	15	sseq1d 3632	. . . . 5 |- ( ( x = X /\ r = R ) -> ( x C_ A <-> X C_ A ) )
17		simpr 477	. . . . . 6 |- ( ( x = X /\ r = R ) -> r = R )
18	15	sqxpeqd 5141	. . . . . 6 |- ( ( x = X /\ r = R ) -> ( x X. x ) = ( X X. X ) )
19	17, 18	sseq12d 3634	. . . . 5 |- ( ( x = X /\ r = R ) -> ( r C_ ( x X. x ) <-> R C_ ( X X. X ) ) )
20	16, 19	anbi12d 747	. . . 4 |- ( ( x = X /\ r = R ) -> ( ( x C_ A /\ r C_ ( x X. x ) ) <-> ( X C_ A /\ R C_ ( X X. X ) ) ) )
21		weeq2 5103	. . . . . 6 |- ( x = X -> ( r We x <-> r We X ) )
22		weeq1 5102	. . . . . 6 |- ( r = R -> ( r We X <-> R We X ) )
23	21, 22	sylan9bb 736	. . . . 5 |- ( ( x = X /\ r = R ) -> ( r We x <-> R We X ) )
24	17	cnveqd 5298	. . . . . . . 8 |- ( ( x = X /\ r = R ) -> `' r = `' R )
25	24	imaeq1d 5465	. . . . . . 7 |- ( ( x = X /\ r = R ) -> ( `' r " { y } ) = ( `' R " { y } ) )
26	17	ineq1d 3813	. . . . . . . . 9 |- ( ( x = X /\ r = R ) -> ( r i^i ( u X. u ) ) = ( R i^i ( u X. u ) ) )
27	26	oveq2d 6666	. . . . . . . 8 |- ( ( x = X /\ r = R ) -> ( u F ( r i^i ( u X. u ) ) ) = ( u F ( R i^i ( u X. u ) ) ) )
28	27	eqeq1d 2624	. . . . . . 7 |- ( ( x = X /\ r = R ) -> ( ( u F ( r i^i ( u X. u ) ) ) = y <-> ( u F ( R i^i ( u X. u ) ) ) = y ) )
29	25, 28	sbceqbid 3442	. . . . . 6 |- ( ( x = X /\ r = R ) -> ( [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y <-> [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) )
30	15, 29	raleqbidv 3152	. . . . 5 |- ( ( x = X /\ r = R ) -> ( A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y <-> A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) )
31	23, 30	anbi12d 747	. . . 4 |- ( ( x = X /\ r = R ) -> ( ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) <-> ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) )
32	20, 31	anbi12d 747	. . 3 |- ( ( x = X /\ r = R ) -> ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) <-> ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) )
33	32, 1	brabga 4989	. 2 |- ( ( X e. _V /\ R e. _V ) -> ( X W R <-> ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) )
34	5, 14, 33	pm5.21nd 941	1 |- ( ph -> ( X W R <-> ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   "cima 5117   Rel wrel 5119  (class class class)co 6650

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  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-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  fpwwe2lem3  9455  fpwwe2lem6  9457  fpwwe2lem7  9458  fpwwe2lem9  9460  fpwwe2lem11  9462  fpwwe2lem12  9463  fpwwe2lem13  9464  fpwwe2  9465  canthwelem  9472  pwfseqlem4  9484