Theorem ovmpod

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014)

	Ref	Expression
Hypotheses	ovmpod.1	|- ( ph -> F = ( x e. C , y e. D |-> R ) )
	ovmpod.2	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
	ovmpod.3	|- ( ph -> A e. C )
	ovmpod.4	|- ( ph -> B e. D )
	ovmpod.5	|- ( ph -> S e. X )
Assertion	ovmpod	|- ( ph -> ( A F B ) = S )

Proof

Step	Hyp	Ref	Expression
1		ovmpod.1	|- ( ph -> F = ( x e. C , y e. D |-> R ) )
2		ovmpod.2	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
3		ovmpod.3	|- ( ph -> A e. C )
4		ovmpod.4	|- ( ph -> B e. D )
5		ovmpod.5	|- ( ph -> S e. X )
6		eqidd	|- ( ( ph /\ x = A ) -> D = D )
7	1 2 6 3 4 5	ovmpodx	|- ( ph -> ( A F B ) = S )