Theorem resmptd

Description: Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypothesis	resmptd.b	$$⊢\phi \to B\subseteq A$$
Assertion	resmptd	$$⊢\phi \to {\left(x\in A⟼C\right)\upharpoonright }_B=\left(x\in B⟼C\right)$$

Proof

Step	Hyp	Ref	Expression
1		resmptd.b	$$⊢\phi \to B\subseteq A$$
2		resmpt	$$⊢B\subseteq A\to {\left(x\in A⟼C\right)\upharpoonright }_B=\left(x\in B⟼C\right)$$
3	1 2	syl	$$⊢\phi \to {\left(x\in A⟼C\right)\upharpoonright }_B=\left(x\in B⟼C\right)$$