Theorem ssres

Description: Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994)

	Ref	Expression
Assertion	ssres	$$⊢A\subseteq B\to {A\upharpoonright }_C\subseteq {B\upharpoonright }_C$$

Proof

Step	Hyp	Ref	Expression
1		ssrin	$$⊢A\subseteq B\to A\cap \left(C\times \mathrm{V}\right)\subseteq B\cap \left(C\times \mathrm{V}\right)$$
2		df-res	$$⊢{A\upharpoonright }_C=A\cap \left(C\times \mathrm{V}\right)$$
3		df-res	$$⊢{B\upharpoonright }_C=B\cap \left(C\times \mathrm{V}\right)$$
4	1 2 3	3sstr4g	$$⊢A\subseteq B\to {A\upharpoonright }_C\subseteq {B\upharpoonright }_C$$