Theorem funfn

Description: A class is a function if and only if it is a function on its domain. (Contributed by NM, 13-Aug-2004)

	Ref	Expression
Assertion	funfn	$$⊢\mathrm{Fun}A\leftrightarrow AFn\mathrm{dom}A$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{dom}A=\mathrm{dom}A$$
2	1	biantru	$$⊢\mathrm{Fun}A\leftrightarrow \left(\mathrm{Fun}A\wedge \mathrm{dom}A=\mathrm{dom}A\right)$$
3		df-fn	$$⊢AFn\mathrm{dom}A\leftrightarrow \left(\mathrm{Fun}A\wedge \mathrm{dom}A=\mathrm{dom}A\right)$$
4	2 3	bitr4i	$$⊢\mathrm{Fun}A\leftrightarrow AFn\mathrm{dom}A$$