Theorem fovrnd

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

	Ref	Expression
Hypotheses	fovrnd.1	$$⊢\phi \to F:R\times S⟶C$$
	fovrnd.2	$$⊢\phi \to A\in R$$
	fovrnd.3	$$⊢\phi \to B\in S$$
Assertion	fovrnd	$$⊢\phi \to AFB\in C$$

Proof

Step	Hyp	Ref	Expression
1		fovrnd.1	$$⊢\phi \to F:R\times S⟶C$$
2		fovrnd.2	$$⊢\phi \to A\in R$$
3		fovrnd.3	$$⊢\phi \to B\in S$$
4		fovrn	$$⊢\left(F:R\times S⟶C\wedge A\in R\wedge B\in S\right)\to AFB\in C$$
5	1 2 3 4	syl3anc	$$⊢\phi \to AFB\in C$$