Theorem ovmpt2df 5632

Description: Alternate deduction version of ovmpt2 5636, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
ovmpt2df.1	|- ( ph -> A e. C )
ovmpt2df.2	|- ( ( ph /\ x = A ) -> B e. D )
ovmpt2df.3	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
ovmpt2df.4	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
ovmpt2df.5	|- F/_ x F
ovmpt2df.6	|- F/ x ps
ovmpt2df.7	|- F/_ y F
ovmpt2df.8	|- F/ y ps

Assertion

Ref	Expression
ovmpt2df	|- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Distinct variable groups:   x, y, A   y, B   ph, x, y

Allowed substitution hints:   ps( x, y)   B( x)   C( x, y)   D( x, y)   R( x, y)   F( x, y)   V( x, y)

Proof of Theorem ovmpt2df

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 |- F/ x ph
2		ovmpt2df.5	. . . 4 |- F/_ x F
3		nfmpt21 5571	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
4	2, 3	nfeq 2185	. . 3 |- F/ x F = ( x e. C , y e. D |-> R )
5		ovmpt2df.6	. . 3 |- F/ x ps
6	4, 5	nfim 1464	. 2 |- F/ x ( F = ( x e. C , y e. D |-> R ) -> ps )
7		ovmpt2df.1	. . . 4 |- ( ph -> A e. C )
8		elex 2566	. . . 4 |- ( A e. C -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2561	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 127	. 2 |- ( ph -> E. x x = A )
12		ovmpt2df.2	. . . . 5 |- ( ( ph /\ x = A ) -> B e. D )
13		elex 2566	. . . . 5 |- ( B e. D -> B e. _V )
14	12, 13	syl 14	. . . 4 |- ( ( ph /\ x = A ) -> B e. _V )
15		isset 2561	. . . 4 |- ( B e. _V <-> E. y y = B )
16	14, 15	sylib 127	. . 3 |- ( ( ph /\ x = A ) -> E. y y = B )
17		nfv 1421	. . . 4 |- F/ y ( ph /\ x = A )
18		ovmpt2df.7	. . . . . 6 |- F/_ y F
19		nfmpt22 5572	. . . . . 6 |- F/_ y ( x e. C , y e. D |-> R )
20	18, 19	nfeq 2185	. . . . 5 |- F/ y F = ( x e. C , y e. D |-> R )
21		ovmpt2df.8	. . . . 5 |- F/ y ps
22	20, 21	nfim 1464	. . . 4 |- F/ y ( F = ( x e. C , y e. D |-> R ) -> ps )
23		oveq 5518	. . . . . 6 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
24		simprl 483	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x = A )
25		simprr 484	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y = B )
26	24, 25	oveq12d 5530	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x ( x e. C , y e. D |-> R ) y ) = ( A ( x e. C , y e. D |-> R ) B ) )
27	7	adantr 261	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> A e. C )
28	24, 27	eqeltrd 2114	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x e. C )
29	12	adantrr 448	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> B e. D )
30	25, 29	eqeltrd 2114	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y e. D )
31		ovmpt2df.3	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
32		eqid 2040	. . . . . . . . . . 11 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
33	32	ovmpt4g 5623	. . . . . . . . . 10 |- ( ( x e. C /\ y e. D /\ R e. V ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
34	28, 30, 31, 33	syl3anc 1135	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
35	26, 34	eqtr3d 2074	. . . . . . . 8 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( A ( x e. C , y e. D |-> R ) B ) = R )
36	35	eqeq2d 2051	. . . . . . 7 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) <-> ( A F B ) = R ) )
37		ovmpt2df.4	. . . . . . 7 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
38	36, 37	sylbid 139	. . . . . 6 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) -> ps ) )
39	23, 38	syl5 28	. . . . 5 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )
40	39	expr 357	. . . 4 |- ( ( ph /\ x = A ) -> ( y = B -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) )
41	17, 22, 40	exlimd 1488	. . 3 |- ( ( ph /\ x = A ) -> ( E. y y = B -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) )
42	16, 41	mpd 13	. 2 |- ( ( ph /\ x = A ) -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )
43	1, 6, 11, 42	exlimdd 1752	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   F/wnf 1349   E.wex 1381   e. wcel 1393   F/_wnfc 2165   _Vcvv 2557  (class class class)co 5512   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  ovmpt2dv  5633  ovmpt2dv2  5634