Theorem ofmresval

Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014)

	Ref	Expression
Hypotheses	ofmresval.f	$$⊢\phi \to F\in A$$
	ofmresval.g	$$⊢\phi \to G\in B$$
Assertion	ofmresval	$$⊢\phi \to F\left({{\circ }_{f}R\upharpoonright }_{\left(A\times B\right)}\right)G=F{R}_{f}G$$

Proof

Step	Hyp	Ref	Expression
1		ofmresval.f	$$⊢\phi \to F\in A$$
2		ofmresval.g	$$⊢\phi \to G\in B$$
3		ovres	$$⊢\left(F\in A\wedge G\in B\right)\to F\left({{\circ }_{f}R\upharpoonright }_{\left(A\times B\right)}\right)G=F{R}_{f}G$$
4	1 2 3	syl2anc	$$⊢\phi \to F\left({{\circ }_{f}R\upharpoonright }_{\left(A\times B\right)}\right)G=F{R}_{f}G$$