Theorem ovmpodv 5943

Description: Alternate deduction version of ovmpo 5946, 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 ) )

Assertion

Ref	Expression
ovmpodv	|- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Distinct variable groups:   x, y, A   y, B   ph, x, y   x, F, y   ps, x, y

Allowed substitution hints:   B( x)   C( x, y)   D( x, y)   R( x, y)   V( x, y)

Proof of Theorem ovmpodv

Step	Hyp	Ref	Expression
1		ovmpodf.1	. 2 |- ( ph -> A e. C )
2		ovmpodf.2	. 2 |- ( ( ph /\ x = A ) -> B e. D )
3		ovmpodf.3	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
4		ovmpodf.4	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
5		nfcv 2296	. 2 |- F/_ x F
6		nfv 1505	. 2 |- F/ x ps
7		nfcv 2296	. 2 |- F/_ y F
8		nfv 1505	. 2 |- F/ y ps
9	1, 2, 3, 4, 5, 6, 7, 8	ovmpodf 5942	1 |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 2125  (class class class)co 5814   e. cmpo 5816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-setind 4490

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819

This theorem is referenced by: (None)