Theorem ovmpox

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

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

Proof

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