Theorem dmresv

Description: The domain of a universal restriction. (Contributed by NM, 14-May-2008)

	Ref	Expression
Assertion	dmresv	$$⊢\mathrm{dom}\left({A\upharpoonright }_{\mathrm{V}}\right)=\mathrm{dom}A$$

Proof

Step	Hyp	Ref	Expression
1		dmres	$$⊢\mathrm{dom}\left({A\upharpoonright }_{\mathrm{V}}\right)=\mathrm{V}\cap \mathrm{dom}A$$
2		incom	$$⊢\mathrm{V}\cap \mathrm{dom}A=\mathrm{dom}A\cap \mathrm{V}$$
3		inv1	$$⊢\mathrm{dom}A\cap \mathrm{V}=\mathrm{dom}A$$
4	1 2 3	3eqtri	$$⊢\mathrm{dom}\left({A\upharpoonright }_{\mathrm{V}}\right)=\mathrm{dom}A$$