Theorem ovmpt4g

Description: Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 .) (Contributed by NM, 21-Feb-2004) (Revised by Mario Carneiro, 1-Sep-2015)

	Ref	Expression
Hypothesis	ovmpt4g.3	|- F = ( x e. A , y e. B |-> C )
Assertion	ovmpt4g	|- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )

Proof

Step	Hyp	Ref	Expression
1		ovmpt4g.3	|- F = ( x e. A , y e. B |-> C )
2		elisset	|- ( C e. V -> E. z z = C )
3		moeq	|- E* z z = C
4	3	a1i	|- ( ( x e. A /\ y e. B ) -> E* z z = C )
5		df-mpo	|- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
6	1 5	eqtri	|- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7	4 6	ovidi	|- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = z ) )
8		eqeq2	|- ( z = C -> ( ( x F y ) = z <-> ( x F y ) = C ) )
9	7 8	mpbidi	|- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = C ) )
10	9	exlimdv	|- ( ( x e. A /\ y e. B ) -> ( E. z z = C -> ( x F y ) = C ) )
11	2 10	syl5	|- ( ( x e. A /\ y e. B ) -> ( C e. V -> ( x F y ) = C ) )
12	11	3impia	|- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )