Theorem mbfdm2

Description: The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014)

	Ref	Expression
Hypotheses	mbfmptcl.1	|- ( ph -> ( x e. A |-> B ) e. MblFn )
	mbfmptcl.2	|- ( ( ph /\ x e. A ) -> B e. V )
Assertion	mbfdm2	|- ( ph -> A e. dom vol )

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	2	ralrimiva	|- ( ph -> A. x e. A B e. V )
4		dmmptg	|- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )
5	3 4	syl	|- ( ph -> dom ( x e. A |-> B ) = A )
6		mbfdm	|- ( ( x e. A |-> B ) e. MblFn -> dom ( x e. A |-> B ) e. dom vol )
7	1 6	syl	|- ( ph -> dom ( x e. A |-> B ) e. dom vol )
8	5 7	eqeltrrd	|- ( ph -> A e. dom vol )