Theorem dmresi

Description: The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004)

	Ref	Expression
Assertion	dmresi	$$⊢\mathrm{dom}\left({\mathrm{I}\upharpoonright }_A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		ssv	$$⊢A\subseteq \mathrm{V}$$
2		dmi	$$⊢\mathrm{dom}\mathrm{I}=\mathrm{V}$$
3	1 2	sseqtrri	$$⊢A\subseteq \mathrm{dom}\mathrm{I}$$
4		ssdmres	$$⊢A\subseteq \mathrm{dom}\mathrm{I}\leftrightarrow \mathrm{dom}\left({\mathrm{I}\upharpoonright }_A\right)=A$$
5	3 4	mpbi	$$⊢\mathrm{dom}\left({\mathrm{I}\upharpoonright }_A\right)=A$$