Theorem ovmpodf 6210

Description: Alternate deduction version of ovmpo 6214, 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 1581	. 2 |- F/ x ph
2		ovmpodf.5	. . . 4 |- F/_ x F
3		nfmpo1 6145	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
4	2, 3	nfeq 2400	. . 3 |- F/ x F = ( x e. C , y e. D |-> R )
5		ovmpodf.6	. . 3 |- F/ x ps
6	4, 5	nfim 1625	. 2 |- F/ x ( F = ( x e. C , y e. D |-> R ) -> ps )
7		ovmpodf.1	. . . 4 |- ( ph -> A e. C )
8		elex 2833	. . . 4 |- ( A e. C -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2828	. . 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 2833	. . . . 5 |- ( B e. D -> B e. _V )
14	12, 13	syl 14	. . . 4 |- ( ( ph /\ x = A ) -> B e. _V )
15		isset 2828	. . . 4 |- ( B e. _V <-> E. y y = B )
16	14, 15	sylib 122	. . 3 |- ( ( ph /\ x = A ) -> E. y y = B )
17		nfv 1581	. . . 4 |- F/ y ( ph /\ x = A )
18		ovmpodf.7	. . . . . 6 |- F/_ y F
19		nfmpo2 6146	. . . . . 6 |- F/_ y ( x e. C , y e. D |-> R )
20	18, 19	nfeq 2400	. . . . 5 |- F/ y F = ( x e. C , y e. D |-> R )
21		ovmpodf.8	. . . . 5 |- F/ y ps
22	20, 21	nfim 1625	. . . 4 |- F/ y ( F = ( x e. C , y e. D |-> R ) -> ps )
23		oveq 6081	. . . . . 6 |- ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = ( A ( x e. C , y e. D |-> R ) B ) )
24		simprl 535	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x = A )
25		simprr 537	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y = B )
26	24, 25	oveq12d 6093	. . . . . . . . 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 2315	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> x e. C )
29	12	adantrr 483	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> B e. D )
30	25, 29	eqeltrd 2315	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y e. D )
31		ovmpodf.3	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
32		eqid 2238	. . . . . . . . . . 11 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
33	32	ovmpt4g 6201	. . . . . . . . . 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 1278	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
35	26, 34	eqtr3d 2273	. . . . . . . 8 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( A ( x e. C , y e. D |-> R ) B ) = R )
36	35	eqeq2d 2250	. . . . . . 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 1650	. . 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 1925	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   F/wnf 1513   E.wex 1545   e. wcel 2209   F/_wnfc 2379   _Vcvv 2821  (class class class)co 6075   e. cmpo 6077

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080

This theorem is referenced by:  ovmpodv  6211  ovmpodv2  6212