Theorem dmresss

Description: The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021) Proof shortened and axiom usage reduced. (Proof shortened by AV, 15-May-2025)

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

Proof

Step	Hyp	Ref	Expression
1		resss	$$⊢{A\upharpoonright }_B\subseteq A$$
2		dmss	$$⊢{A\upharpoonright }_B\subseteq A\to \mathrm{dom}\left({A\upharpoonright }_B\right)\subseteq \mathrm{dom}A$$
3	1 2	ax-mp	$$⊢\mathrm{dom}\left({A\upharpoonright }_B\right)\subseteq \mathrm{dom}A$$