Theorem fnresi

Description: The restricted identity relation is a function on the restricting class. (Contributed by NM, 27-Aug-2004) (Proof shortened by BJ, 27-Dec-2023)

	Ref	Expression
Assertion	fnresi	$$⊢\left({\mathrm{I}\upharpoonright }_A\right)FnA$$

Proof

Step	Hyp	Ref	Expression
1		idfn	$$⊢\mathrm{I}Fn\mathrm{V}$$
2		ssv	$$⊢A\subseteq \mathrm{V}$$
3		fnssres	$$⊢\left(\mathrm{I}Fn\mathrm{V}\wedge A\subseteq \mathrm{V}\right)\to \left({\mathrm{I}\upharpoonright }_A\right)FnA$$
4	1 2 3	mp2an	$$⊢\left({\mathrm{I}\upharpoonright }_A\right)FnA$$