Metamath Proof Explorer

Theorem dffn4

Description: A function maps onto its range. (Contributed by NM, 10-May-1998)

		Ref	Expression
	Assertion	dffn4	$⊢ F Fn A \leftrightarrow F : A \underset{onto}{⟶} ran F$

Step	Hyp	Ref	Expression
1		eqid	$⊢ ran F = ran F$
2	1	biantru	$⊢ F Fn A \leftrightarrow (F Fn A \land ran F = ran F)$
3		df-fo	$⊢ F : A \underset{onto}{⟶} ran F \leftrightarrow (F Fn A \land ran F = ran F)$
4	2 3	bitr4i	$⊢ F Fn A \leftrightarrow F : A \underset{onto}{⟶} ran F$