Theorem ovmpt2df 5792

Description: Alternate deduction version of ovmpt2 5796, 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 1467	. 2 |- F/ x ph
2		ovmpt2df.5	. . . 4 |- F/_ x F
3		nfmpt21 5731	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
4	2, 3	nfeq 2237	. . 3 |- F/ x F = ( x e. C , y e. D |-> R )
5		ovmpt2df.6	. . 3 |- F/ x ps
6	4, 5	nfim 1510	. 2 |- F/ x ( F = ( x e. C , y e. D |-> R ) -> ps )
7		ovmpt2df.1	. . . 4 |- ( ph -> A e. C )
8		elex 2633	. . . 4 |- ( A e. C -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2628	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 121	. 2 |- ( ph -> E. x x = A )
12		ovmpt2df.2	. . . . 5 |- ( ( ph /\ x = A ) -> B e. D )
13		elex 2633	. . . . 5 |- ( B e. D -> B e. _V )
14	12, 13	syl 14	. . . 4 |- ( ( ph /\ x = A ) -> B e. _V )
15		isset 2628	. . . 4 |- ( B e. _V <-> E. y y = B )
16	14, 15	sylib 121	. . 3 |- ( ( ph /\ x = A ) -> E. y y = B )
17		nfv 1467	. . . 4 |- F/ y ( ph /\ x = A )
18		ovmpt2df.7	. . . . . 6 |- F/_ y F
19		nfmpt22 5732	. . . . . 6 |- F/_ y ( x e. C , y e. D |-> R )
20	18, 19	nfeq 2237	. . . . 5 |- F/ y F = ( x e. C , y e. D |-> R )
21		ovmpt2df.8	. . . . 5 |- F/ y ps
22	20, 21	nfim 1510	. . . 4 |- F/ y ( F = ( x e. C , y e. D |-> R ) -> ps )
23		oveq 5674	. . . . . 6 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
24		simprl 499	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x = A )
25		simprr 500	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y = B )
26	24, 25	oveq12d 5686	. . . . . . . . 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 271	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> A e. C )
28	24, 27	eqeltrd 2165	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x e. C )
29	12	adantrr 464	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> B e. D )
30	25, 29	eqeltrd 2165	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y e. D )
31		ovmpt2df.3	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
32		eqid 2089	. . . . . . . . . . 11 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
33	32	ovmpt4g 5783	. . . . . . . . . 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 1175	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
35	26, 34	eqtr3d 2123	. . . . . . . 8 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( A ( x e. C , y e. D |-> R ) B ) = R )
36	35	eqeq2d 2100	. . . . . . 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 149	. . . . . 6 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) -> ps ) )
39	23, 38	syl5 32	. . . . 5 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )
40	39	expr 368	. . . 4 |- ( ( ph /\ x = A ) -> ( y = B -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) )
41	17, 22, 40	exlimd 1534	. . 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 1801	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   F/wnf 1395   E.wex 1427   e. wcel 1439   F/_wnfc 2216   _Vcvv 2622  (class class class)co 5668   |-> cmpt2 5670

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-setind 4368

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673

This theorem is referenced by:  ovmpt2dv  5793  ovmpt2dv2  5794