Theorem ovmpodf 6100

Description: Alternate deduction version of ovmpo 6104, 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 1552	. 2 |- F/ x ph
2		ovmpodf.5	. . . 4 |- F/_ x F
3		nfmpo1 6035	. . . 4 |- F/_ x ( x e. C , y e. D |-> R )
4	2, 3	nfeq 2358	. . 3 |- F/ x F = ( x e. C , y e. D |-> R )
5		ovmpodf.6	. . 3 |- F/ x ps
6	4, 5	nfim 1596	. 2 |- F/ x ( F = ( x e. C , y e. D |-> R ) -> ps )
7		ovmpodf.1	. . . 4 |- ( ph -> A e. C )
8		elex 2788	. . . 4 |- ( A e. C -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2783	. . 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 2788	. . . . 5 |- ( B e. D -> B e. _V )
14	12, 13	syl 14	. . . 4 |- ( ( ph /\ x = A ) -> B e. _V )
15		isset 2783	. . . 4 |- ( B e. _V <-> E. y y = B )
16	14, 15	sylib 122	. . 3 |- ( ( ph /\ x = A ) -> E. y y = B )
17		nfv 1552	. . . 4 |- F/ y ( ph /\ x = A )
18		ovmpodf.7	. . . . . 6 |- F/_ y F
19		nfmpo2 6036	. . . . . 6 |- F/_ y ( x e. C , y e. D |-> R )
20	18, 19	nfeq 2358	. . . . 5 |- F/ y F = ( x e. C , y e. D |-> R )
21		ovmpodf.8	. . . . 5 |- F/ y ps
22	20, 21	nfim 1596	. . . 4 |- F/ y ( F = ( x e. C , y e. D |-> R ) -> ps )
23		oveq 5973	. . . . . 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 5985	. . . . . . . . 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 2284	. . . . . . . . . 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 2284	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> y e. D )
31		ovmpodf.3	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
32		eqid 2207	. . . . . . . . . . 11 |- ( x e. C , y e. D |-> R ) = ( x e. C , y e. D |-> R )
33	32	ovmpt4g 6091	. . . . . . . . . 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 1250	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x ( x e. C , y e. D |-> R ) y ) = R )
35	26, 34	eqtr3d 2242	. . . . . . . 8 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( A ( x e. C , y e. D |-> R ) B ) = R )
36	35	eqeq2d 2219	. . . . . . 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 1621	. . 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 1896	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   F/wnf 1484   E.wex 1516   e. wcel 2178   F/_wnfc 2337   _Vcvv 2776  (class class class)co 5967   e. cmpo 5969

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972

This theorem is referenced by:  ovmpodv  6101  ovmpodv2  6102