Theorem ovmpo

Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995) (Revised by David Abernethy, 19-Jun-2012)

	Ref	Expression
Hypotheses	ovmpog.1	|- ( x = A -> R = G )
	ovmpog.2	|- ( y = B -> G = S )
	ovmpog.3	|- F = ( x e. C , y e. D |-> R )
	ovmpo.4	|- S e. _V
Assertion	ovmpo	|- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )

Proof

Step	Hyp	Ref	Expression
1		ovmpog.1	|- ( x = A -> R = G )
2		ovmpog.2	|- ( y = B -> G = S )
3		ovmpog.3	|- F = ( x e. C , y e. D |-> R )
4		ovmpo.4	|- S e. _V
5	1 2 3	ovmpog	|- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
6	4 5	mp3an3	|- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )