Theorem ovmpoga

Description: Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	ovmpoga.1	|- ( ( x = A /\ y = B ) -> R = S )
	ovmpoga.2	|- F = ( x e. C , y e. D |-> R )
Assertion	ovmpoga	|- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Proof

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