Theorem ssdmres

Description: A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997)

	Ref	Expression
Assertion	ssdmres	$$⊢A\subseteq \mathrm{dom}B\leftrightarrow \mathrm{dom}\left({B\upharpoonright }_A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		dfss2	$$⊢A\subseteq \mathrm{dom}B\leftrightarrow A\cap \mathrm{dom}B=A$$
2		dmres	$$⊢\mathrm{dom}\left({B\upharpoonright }_A\right)=A\cap \mathrm{dom}B$$
3	2	eqeq1i	$$⊢\mathrm{dom}\left({B\upharpoonright }_A\right)=A\leftrightarrow A\cap \mathrm{dom}B=A$$
4	1 3	bitr4i	$$⊢A\subseteq \mathrm{dom}B\leftrightarrow \mathrm{dom}\left({B\upharpoonright }_A\right)=A$$