Theorem ovmpt2df 6792

Description: Alternate deduction version of ovmpt2 6796, 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 1843	. 2 |- F/ x ph
2		ovmpt2df.5	. . . 4 |- F/_ x F
3		nfmpt21 6722	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
4	2, 3	nfeq 2776	. . 3 |- F/ x F = ( x e. C , y e. D |-> R )
5		ovmpt2df.6	. . 3 |- F/ x ps
6	4, 5	nfim 1825	. 2 |- F/ x ( F = ( x e. C , y e. D |-> R ) -> ps )
7		ovmpt2df.1	. . . 4 |- ( ph -> A e. C )
8		elex 3212	. . . 4 |- ( A e. C -> A e. _V )
9	7, 8	syl 17	. . 3 |- ( ph -> A e. _V )
10		isset 3207	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 208	. 2 |- ( ph -> E. x x = A )
12		ovmpt2df.2	. . . . 5 |- ( ( ph /\ x = A ) -> B e. D )
13		elex 3212	. . . . 5 |- ( B e. D -> B e. _V )
14	12, 13	syl 17	. . . 4 |- ( ( ph /\ x = A ) -> B e. _V )
15		isset 3207	. . . 4 |- ( B e. _V <-> E. y y = B )
16	14, 15	sylib 208	. . 3 |- ( ( ph /\ x = A ) -> E. y y = B )
17		nfv 1843	. . . 4 |- F/ y ( ph /\ x = A )
18		ovmpt2df.7	. . . . . 6 |- F/_ y F
19		nfmpt22 6723	. . . . . 6 |- F/_ y ( x e. C , y e. D |-> R )
20	18, 19	nfeq 2776	. . . . 5 |- F/ y F = ( x e. C , y e. D |-> R )
21		ovmpt2df.8	. . . . 5 |- F/ y ps
22	20, 21	nfim 1825	. . . 4 |- F/ y ( F = ( x e. C , y e. D |-> R ) -> ps )
23		oveq 6656	. . . . . 6 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
24		simprl 794	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x = A )
25		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y = B )
26	24, 25	oveq12d 6668	. . . . . . . . 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 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> A e. C )
28	24, 27	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x e. C )
29	12	adantrr 753	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> B e. D )
30	25, 29	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y e. D )
31		ovmpt2df.3	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
32		eqid 2622	. . . . . . . . . . 11 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
33	32	ovmpt4g 6783	. . . . . . . . . 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 1326	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
35	26, 34	eqtr3d 2658	. . . . . . . 8 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( A ( x e. C , y e. D |-> R ) B ) = R )
36	35	eqeq2d 2632	. . . . . . 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 230	. . . . . 6 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) -> ps ) )
39	23, 38	syl5 34	. . . . 5 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )
40	39	expr 643	. . . 4 |- ( ( ph /\ x = A ) -> ( y = B -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) )
41	17, 22, 40	exlimd 2087	. . 3 |- ( ( ph /\ x = A ) -> ( E. y y = B -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) )
42	16, 41	mpd 15	. 2 |- ( ( ph /\ x = A ) -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )
43	1, 6, 11, 42	exlimdd 2088	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200  (class class class)co 6650   |-> cmpt2 6652

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ovmpt2dv  6793  ovmpt2dv2  6794