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Metamath Proof Explorer

Theorem fof

Description: An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994)

		Ref	Expression
	Assertion	fof	$⊢ F : A \underset{onto}{⟶} B \to F : A ⟶ B$

Step	Hyp	Ref	Expression
1		eqimss	$⊢ ran F = B \to ran F \subseteq B$
2	1	anim2i	$⊢ (F Fn A \land ran F = B) \to (F Fn A \land ran F \subseteq B)$
3		df-fo	$⊢ F : A \underset{onto}{⟶} B \leftrightarrow (F Fn A \land ran F = B)$
4		df-f	$⊢ F : A ⟶ B \leftrightarrow (F Fn A \land ran F \subseteq B)$
5	2 3 4	3imtr4i	$⊢ F : A \underset{onto}{⟶} B \to F : A ⟶ B$