Theorem fovrnda

Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016)

	Ref	Expression
Hypothesis	fovrnd.1	$$⊢\phi \to F:R\times S⟶C$$
Assertion	fovrnda	$$⊢\left(\phi \wedge \left(A\in R\wedge B\in S\right)\right)\to AFB\in C$$

Proof

Step	Hyp	Ref	Expression
1		fovrnd.1	$$⊢\phi \to F:R\times S⟶C$$
2		fovrn	$$⊢\left(F:R\times S⟶C\wedge A\in R\wedge B\in S\right)\to AFB\in C$$
3	1 2	syl3an1	$$⊢\left(\phi \wedge A\in R\wedge B\in S\right)\to AFB\in C$$
4	3	3expb	$$⊢\left(\phi \wedge \left(A\in R\wedge B\in S\right)\right)\to AFB\in C$$