Theorem mbfmptcl

Description: Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014)

	Ref	Expression
Hypotheses	mbfmptcl.1	|- ( ph -> ( x e. A |-> B ) e. MblFn )
	mbfmptcl.2	|- ( ( ph /\ x e. A ) -> B e. V )
Assertion	mbfmptcl	|- ( ( ph /\ x e. A ) -> B e. CC )

Proof

Step	Hyp	Ref	Expression
1		mbfmptcl.1	|- ( ph -> ( x e. A |-> B ) e. MblFn )
2		mbfmptcl.2	|- ( ( ph /\ x e. A ) -> B e. V )
3		mbff	|- ( ( x e. A |-> B ) e. MblFn -> ( x e. A |-> B ) : dom ( x e. A |-> B ) --> CC )
4	1 3	syl	|- ( ph -> ( x e. A |-> B ) : dom ( x e. A |-> B ) --> CC )
5	2	ralrimiva	|- ( ph -> A. x e. A B e. V )
6		dmmptg	|- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )
7	5 6	syl	|- ( ph -> dom ( x e. A |-> B ) = A )
8	7	feq2d	|- ( ph -> ( ( x e. A |-> B ) : dom ( x e. A |-> B ) --> CC <-> ( x e. A |-> B ) : A --> CC ) )
9	4 8	mpbid	|- ( ph -> ( x e. A |-> B ) : A --> CC )
10	9	fvmptelrn	|- ( ( ph /\ x e. A ) -> B e. CC )