Theorem fvres

Description: The value of a restricted function. (Contributed by NM, 2-Aug-1994)

	Ref	Expression
Assertion	fvres	$$⊢A\in B\to \left({F\upharpoonright }_B\right)\left(A\right)=F\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2	1	brresi	$$⊢A\left({F\upharpoonright }_B\right)x\leftrightarrow \left(A\in B\wedge AFx\right)$$
3	2	baib	$$⊢A\in B\to \left(A\left({F\upharpoonright }_B\right)x\leftrightarrow AFx\right)$$
4	3	iotabidv	$$⊢A\in B\to \left(\iota x|A\left({F\upharpoonright }_B\right)x\right)=\left(\iota x|AFx\right)$$
5		df-fv	$$⊢\left({F\upharpoonright }_B\right)\left(A\right)=\left(\iota x|A\left({F\upharpoonright }_B\right)x\right)$$
6		df-fv	$$⊢F\left(A\right)=\left(\iota x|AFx\right)$$
7	4 5 6	3eqtr4g	$$⊢A\in B\to \left({F\upharpoonright }_B\right)\left(A\right)=F\left(A\right)$$