Theorem dmresexg

Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995)

	Ref	Expression
Assertion	dmresexg	$$⊢B\in V\to \mathrm{dom}\left({A\upharpoonright }_B\right)\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		dmres	$$⊢\mathrm{dom}\left({A\upharpoonright }_B\right)=B\cap \mathrm{dom}A$$
2		inex1g	$$⊢B\in V\to B\cap \mathrm{dom}A\in \mathrm{V}$$
3	1 2	eqeltrid	$$⊢B\in V\to \mathrm{dom}\left({A\upharpoonright }_B\right)\in \mathrm{V}$$