Theorem fpwwe2lem3 9455

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 )
fpwwe2lem4.4	|- ( ph -> X W R )

Assertion

Ref	Expression
fpwwe2lem3	|- ( ( ph /\ B e. X ) -> ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) = B )

Distinct variable groups:   y, u, B   u, r, x, y, 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)   B( x, r)

Proof of Theorem fpwwe2lem3

Step	Hyp	Ref	Expression
1		fpwwe2lem4.4	. . . . 5 |- ( ph -> X W R )
2		fpwwe2.1	. . . . . 6 |- 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 ) ) }
3		fpwwe2.2	. . . . . 6 |- ( ph -> A e. _V )
4	2, 3	fpwwe2lem2 9454	. . . . 5 |- ( 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 ) ) ) )
5	1, 4	mpbid 222	. . . 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 ) ) )
6	5	simprrd 797	. . 3 |- ( ph -> A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y )
7		sneq 4187	. . . . . 6 |- ( y = B -> { y } = { B } )
8	7	imaeq2d 5466	. . . . 5 |- ( y = B -> ( `' R " { y } ) = ( `' R " { B } ) )
9		eqeq2 2633	. . . . 5 |- ( y = B -> ( ( u F ( R i^i ( u X. u ) ) ) = y <-> ( u F ( R i^i ( u X. u ) ) ) = B ) )
10	8, 9	sbceqbid 3442	. . . 4 |- ( y = B -> ( [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y <-> [. ( `' R " { B } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = B ) )
11	10	rspccva 3308	. . 3 |- ( ( A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y /\ B e. X ) -> [. ( `' R " { B } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = B )
12	6, 11	sylan 488	. 2 |- ( ( ph /\ B e. X ) -> [. ( `' R " { B } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = B )
13		cnvimass 5485	. . . . 5 |- ( `' R " { B } ) C_ dom R
14	2	relopabi 5245	. . . . . . 7 |- Rel W
15	14	brrelex2i 5159	. . . . . 6 |- ( X W R -> R e. _V )
16		dmexg 7097	. . . . . 6 |- ( R e. _V -> dom R e. _V )
17	1, 15, 16	3syl 18	. . . . 5 |- ( ph -> dom R e. _V )
18		ssexg 4804	. . . . 5 |- ( ( ( `' R " { B } ) C_ dom R /\ dom R e. _V ) -> ( `' R " { B } ) e. _V )
19	13, 17, 18	sylancr 695	. . . 4 |- ( ph -> ( `' R " { B } ) e. _V )
20		id 22	. . . . . . 7 |- ( u = ( `' R " { B } ) -> u = ( `' R " { B } ) )
21	20	sqxpeqd 5141	. . . . . . . 8 |- ( u = ( `' R " { B } ) -> ( u X. u ) = ( ( `' R " { B } ) X. ( `' R " { B } ) ) )
22	21	ineq2d 3814	. . . . . . 7 |- ( u = ( `' R " { B } ) -> ( R i^i ( u X. u ) ) = ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) )
23	20, 22	oveq12d 6668	. . . . . 6 |- ( u = ( `' R " { B } ) -> ( u F ( R i^i ( u X. u ) ) ) = ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) )
24	23	eqeq1d 2624	. . . . 5 |- ( u = ( `' R " { B } ) -> ( ( u F ( R i^i ( u X. u ) ) ) = B <-> ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) = B ) )
25	24	sbcieg 3468	. . . 4 |- ( ( `' R " { B } ) e. _V -> ( [. ( `' R " { B } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = B <-> ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) = B ) )
26	19, 25	syl 17	. . 3 |- ( ph -> ( [. ( `' R " { B } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = B <-> ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) = B ) )
27	26	adantr 481	. 2 |- ( ( ph /\ B e. X ) -> ( [. ( `' R " { B } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = B <-> ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) = B ) )
28	12, 27	mpbid 222	1 |- ( ( ph /\ B e. X ) -> ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) = B )

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   dom cdm 5114   "cima 5117  (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:  fpwwe2lem8  9459  fpwwe2lem12  9463  fpwwe2lem13  9464  fpwwe2  9465  canthwelem  9472  pwfseqlem4  9484