Theorem ffrnbd

Description: A function maps to its range iff the range is a subset of its codomain. Generalization of ffrn . (Contributed by AV, 20-Sep-2024)

	Ref	Expression
Hypothesis	ffrnbd.r	$$⊢\phi \to \mathrm{ran}F\subseteq B$$
Assertion	ffrnbd	$$⊢\phi \to \left(F:A⟶B\leftrightarrow F:A⟶\mathrm{ran}F\right)$$

Proof

Step	Hyp	Ref	Expression
1		ffrnbd.r	$$⊢\phi \to \mathrm{ran}F\subseteq B$$
2		ffrnb	$$⊢F:A⟶B\leftrightarrow \left(F:A⟶\mathrm{ran}F\wedge \mathrm{ran}F\subseteq B\right)$$
3	1	biantrud	$$⊢\phi \to \left(F:A⟶\mathrm{ran}F\leftrightarrow \left(F:A⟶\mathrm{ran}F\wedge \mathrm{ran}F\subseteq B\right)\right)$$
4	2 3	bitr4id	$$⊢\phi \to \left(F:A⟶B\leftrightarrow F:A⟶\mathrm{ran}F\right)$$