Theorem ovmpodf 6054

Description: Alternate deduction version of ovmpo 6058, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
ovmpodf	|- ( 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 ovmpodf

Step	Hyp	Ref	Expression
1		nfv 1542	. 2 |- F/ x ph
2		ovmpodf.5	. . . 4 |- F/_ x F
3		nfmpo1 5989	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
4	2, 3	nfeq 2347	. . 3 |- F/ x F = ( x e. C , y e. D |-> R )
5		ovmpodf.6	. . 3 |- F/ x ps
6	4, 5	nfim 1586	. 2 |- F/ x ( F = ( x e. C , y e. D |-> R ) -> ps )
7		ovmpodf.1	. . . 4 |- ( ph -> A e. C )
8		elex 2774	. . . 4 |- ( A e. C -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2769	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 122	. 2 |- ( ph -> E. x x = A )
12		ovmpodf.2	. . . . 5 |- ( ( ph /\ x = A ) -> B e. D )
13		elex 2774	. . . . 5 |- ( B e. D -> B e. _V )
14	12, 13	syl 14	. . . 4 |- ( ( ph /\ x = A ) -> B e. _V )
15		isset 2769	. . . 4 |- ( B e. _V <-> E. y y = B )
16	14, 15	sylib 122	. . 3 |- ( ( ph /\ x = A ) -> E. y y = B )
17		nfv 1542	. . . 4 |- F/ y ( ph /\ x = A )
18		ovmpodf.7	. . . . . 6 |- F/_ y F
19		nfmpo2 5990	. . . . . 6 |- F/_ y ( x e. C , y e. D |-> R )
20	18, 19	nfeq 2347	. . . . 5 |- F/ y F = ( x e. C , y e. D |-> R )
21		ovmpodf.8	. . . . 5 |- F/ y ps
22	20, 21	nfim 1586	. . . 4 |- F/ y ( F = ( x e. C , y e. D |-> R ) -> ps )
23		oveq 5928	. . . . . 6 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
24		simprl 529	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x = A )
25		simprr 531	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y = B )
26	24, 25	oveq12d 5940	. . . . . . . . 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 276	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> A e. C )
28	24, 27	eqeltrd 2273	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x e. C )
29	12	adantrr 479	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> B e. D )
30	25, 29	eqeltrd 2273	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y e. D )
31		ovmpodf.3	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
32		eqid 2196	. . . . . . . . . . 11 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
33	32	ovmpt4g 6045	. . . . . . . . . 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 1249	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
35	26, 34	eqtr3d 2231	. . . . . . . 8 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( A ( x e. C , y e. D |-> R ) B ) = R )
36	35	eqeq2d 2208	. . . . . . 7 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) <-> ( A F B ) = R ) )
37		ovmpodf.4	. . . . . . 7 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
38	36, 37	sylbid 150	. . . . . 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 375	. . . 4 |- ( ( ph /\ x = A ) -> ( y = B -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) )
41	17, 22, 40	exlimd 1611	. . 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 1886	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   F/wnf 1474   E.wex 1506   e. wcel 2167   F/_wnfc 2326   _Vcvv 2763  (class class class)co 5922   e. cmpo 5924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927

This theorem is referenced by:  ovmpodv  6055  ovmpodv2  6056