Theorem resmpt

Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013)

	Ref	Expression
Assertion	resmpt	$$⊢B\subseteq A\to {\left(x\in A⟼C\right)\upharpoonright }_B=\left(x\in B⟼C\right)$$

Proof

Step	Hyp	Ref	Expression
1		resopab2	$$⊢B\subseteq A\to {\left\{⟨x,y⟩|\left(x\in A\wedge y=C\right)\right\}\upharpoonright }_B=\left\{⟨x,y⟩|\left(x\in B\wedge y=C\right)\right\}$$
2		df-mpt	$$⊢\left(x\in A⟼C\right)=\left\{⟨x,y⟩|\left(x\in A\wedge y=C\right)\right\}$$
3	2	reseq1i	$$⊢{\left(x\in A⟼C\right)\upharpoonright }_B={\left\{⟨x,y⟩|\left(x\in A\wedge y=C\right)\right\}\upharpoonright }_B$$
4		df-mpt	$$⊢\left(x\in B⟼C\right)=\left\{⟨x,y⟩|\left(x\in B\wedge y=C\right)\right\}$$
5	1 3 4	3eqtr4g	$$⊢B\subseteq A\to {\left(x\in A⟼C\right)\upharpoonright }_B=\left(x\in B⟼C\right)$$