Theorem ovmpox 5899

Description: The value of an operation class abstraction. Variant of ovmpoga 5900 which does not require D and x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Hypotheses

Ref	Expression
ovmpox.1	|- ( ( x = A /\ y = B ) -> R = S )
ovmpox.2	|- ( x = A -> D = L )
ovmpox.3	|- F = ( x e. C , y e. D |-> R )

Assertion

Ref	Expression
ovmpox	|- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, L, y   x, S, y

Allowed substitution hints:   D( x, y)   R( x, y)   F( x, y)   H( x, y)

Proof of Theorem ovmpox

Step	Hyp	Ref	Expression
1		elex 2697	. 2 |- ( S e. H -> S e. _V )
2		ovmpox.3	. . . 4 |- F = ( x e. C , y e. D |-> R )
3	2	a1i 9	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> F = ( x e. C , y e. D |-> R ) )
4		ovmpox.1	. . . 4 |- ( ( x = A /\ y = B ) -> R = S )
5	4	adantl 275	. . 3 |- ( ( ( A e. C /\ B e. L /\ S e. _V ) /\ ( x = A /\ y = B ) ) -> R = S )
6		ovmpox.2	. . . 4 |- ( x = A -> D = L )
7	6	adantl 275	. . 3 |- ( ( ( A e. C /\ B e. L /\ S e. _V ) /\ x = A ) -> D = L )
8		simp1 981	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> A e. C )
9		simp2 982	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> B e. L )
10		simp3 983	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> S e. _V )
11	3, 5, 7, 8, 9, 10	ovmpodx 5897	. 2 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> ( A F B ) = S )
12	1, 11	syl3an3 1251	1 |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   _Vcvv 2686  (class class class)co 5774   e. cmpo 5776

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  reldvg  12817