Theorem dmresss

Description: The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Assertion	dmresss	$$⊢\mathrm{dom}\left({A\upharpoonright }_B\right)\subseteq \mathrm{dom}A$$

Proof

Step	Hyp	Ref	Expression
1		dmres	$$⊢\mathrm{dom}\left({A\upharpoonright }_B\right)=B\cap \mathrm{dom}A$$
2		inss2	$$⊢B\cap \mathrm{dom}A\subseteq \mathrm{dom}A$$
3	1 2	eqsstri	$$⊢\mathrm{dom}\left({A\upharpoonright }_B\right)\subseteq \mathrm{dom}A$$