Theorem fpwwe2lem5 9456

Description: Lemma for fpwwe2 9465. (Contributed by Mario Carneiro, 15-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 )
fpwwe2.3	|- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )

Assertion

Ref	Expression
fpwwe2lem5	|- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> ( X F R ) e. A )

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 fpwwe2lem5

Step	Hyp	Ref	Expression
1		fpwwe2.2	. . . . 5 |- ( ph -> A e. _V )
2	1	adantr 481	. . . 4 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> A e. _V )
3		simpr1 1067	. . . 4 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> X C_ A )
4	2, 3	ssexd 4805	. . 3 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> X e. _V )
5		xpexg 6960	. . . . 5 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
6	4, 4, 5	syl2anc 693	. . . 4 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> ( X X. X ) e. _V )
7		simpr2 1068	. . . 4 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> R C_ ( X X. X ) )
8	6, 7	ssexd 4805	. . 3 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> R e. _V )
9	4, 8	jca 554	. 2 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> ( X e. _V /\ R e. _V ) )
10		sseq1 3626	. . . . . 6 |- ( x = X -> ( x C_ A <-> X C_ A ) )
11		xpeq12 5134	. . . . . . . 8 |- ( ( x = X /\ x = X ) -> ( x X. x ) = ( X X. X ) )
12	11	anidms 677	. . . . . . 7 |- ( x = X -> ( x X. x ) = ( X X. X ) )
13	12	sseq2d 3633	. . . . . 6 |- ( x = X -> ( r C_ ( x X. x ) <-> r C_ ( X X. X ) ) )
14		weeq2 5103	. . . . . 6 |- ( x = X -> ( r We x <-> r We X ) )
15	10, 13, 14	3anbi123d 1399	. . . . 5 |- ( x = X -> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) <-> ( X C_ A /\ r C_ ( X X. X ) /\ r We X ) ) )
16	15	anbi2d 740	. . . 4 |- ( x = X -> ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) <-> ( ph /\ ( X C_ A /\ r C_ ( X X. X ) /\ r We X ) ) ) )
17		oveq1 6657	. . . . 5 |- ( x = X -> ( x F r ) = ( X F r ) )
18	17	eleq1d 2686	. . . 4 |- ( x = X -> ( ( x F r ) e. A <-> ( X F r ) e. A ) )
19	16, 18	imbi12d 334	. . 3 |- ( x = X -> ( ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A ) <-> ( ( ph /\ ( X C_ A /\ r C_ ( X X. X ) /\ r We X ) ) -> ( X F r ) e. A ) ) )
20		sseq1 3626	. . . . . 6 |- ( r = R -> ( r C_ ( X X. X ) <-> R C_ ( X X. X ) ) )
21		weeq1 5102	. . . . . 6 |- ( r = R -> ( r We X <-> R We X ) )
22	20, 21	3anbi23d 1402	. . . . 5 |- ( r = R -> ( ( X C_ A /\ r C_ ( X X. X ) /\ r We X ) <-> ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) )
23	22	anbi2d 740	. . . 4 |- ( r = R -> ( ( ph /\ ( X C_ A /\ r C_ ( X X. X ) /\ r We X ) ) <-> ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) ) )
24		oveq2 6658	. . . . 5 |- ( r = R -> ( X F r ) = ( X F R ) )
25	24	eleq1d 2686	. . . 4 |- ( r = R -> ( ( X F r ) e. A <-> ( X F R ) e. A ) )
26	23, 25	imbi12d 334	. . 3 |- ( r = R -> ( ( ( ph /\ ( X C_ A /\ r C_ ( X X. X ) /\ r We X ) ) -> ( X F r ) e. A ) <-> ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> ( X F R ) e. A ) ) )
27		fpwwe2.3	. . 3 |- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
28	19, 26, 27	vtocl2g 3270	. 2 |- ( ( X e. _V /\ R e. _V ) -> ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> ( X F R ) e. A ) )
29	9, 28	mpcom 38	1 |- ( ( ph /\ ( X C_ A /\ R C_ ( X X. X ) /\ R We X ) ) -> ( X F R ) e. A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   {csn 4177   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   "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-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-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-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  fpwwe2lem13  9464