Theorem resdm

Description: A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006)

	Ref	Expression
Assertion	resdm	$$⊢\mathrm{Rel}A\to {A\upharpoonright }_{\mathrm{dom}A}=A$$

Proof

Step	Hyp	Ref	Expression
1		ssid	$$⊢\mathrm{dom}A\subseteq \mathrm{dom}A$$
2		relssres	$$⊢\left(\mathrm{Rel}A\wedge \mathrm{dom}A\subseteq \mathrm{dom}A\right)\to {A\upharpoonright }_{\mathrm{dom}A}=A$$
3	1 2	mpan2	$$⊢\mathrm{Rel}A\to {A\upharpoonright }_{\mathrm{dom}A}=A$$