Theorem fnfvrnss

Description: An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004)

	Ref	Expression
Assertion	fnfvrnss	$$⊢\left(FFnA\wedge \forall x\in AF\left(x\right)\in B\right)\to \mathrm{ran}F\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		ffnfv	$$⊢F:A⟶B\leftrightarrow \left(FFnA\wedge \forall x\in AF\left(x\right)\in B\right)$$
2		frn	$$⊢F:A⟶B\to \mathrm{ran}F\subseteq B$$
3	1 2	sylbir	$$⊢\left(FFnA\wedge \forall x\in AF\left(x\right)\in B\right)\to \mathrm{ran}F\subseteq B$$