Theorem mbfdm2

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

	Ref	Expression
Hypotheses	mbfmptcl.1	$$⊢\phi \to \left(x\in A⟼B\right)\in MblFn$$
	mbfmptcl.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
Assertion	mbfdm2	$$⊢\phi \to A\in \mathrm{dom}vol$$

Proof

Step	Hyp	Ref	Expression
1		mbfmptcl.1	$$⊢\phi \to \left(x\in A⟼B\right)\in MblFn$$
2		mbfmptcl.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
3	2	ralrimiva	$$⊢\phi \to \forall x\in AB\in V$$
4		dmmptg	$$⊢\forall x\in AB\in V\to \mathrm{dom}\left(x\in A⟼B\right)=A$$
5	3 4	syl	$$⊢\phi \to \mathrm{dom}\left(x\in A⟼B\right)=A$$
6		mbfdm	$$⊢\left(x\in A⟼B\right)\in MblFn\to \mathrm{dom}\left(x\in A⟼B\right)\in \mathrm{dom}vol$$
7	1 6	syl	$$⊢\phi \to \mathrm{dom}\left(x\in A⟼B\right)\in \mathrm{dom}vol$$
8	5 7	eqeltrrd	$$⊢\phi \to A\in \mathrm{dom}vol$$