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Metamath Proof Explorer

Theorem f1f1orn

Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004)

		Ref	Expression
	Assertion	f1f1orn	$⊢ F : A \underset{1-1}{⟶} B \to F : A \underset{1-1 onto}{⟶} ran F$

Step	Hyp	Ref	Expression
1		f1fn	$⊢ F : A \underset{1-1}{⟶} B \to F Fn A$
2		df-f1	$⊢ F : A \underset{1-1}{⟶} B \leftrightarrow (F : A ⟶ B \land Fun F^{-1})$
3	2	simprbi	$⊢ F : A \underset{1-1}{⟶} B \to Fun F^{-1}$
4		f1orn	$⊢ F : A \underset{1-1 onto}{⟶} ran F \leftrightarrow (F Fn A \land Fun F^{-1})$
5	1 3 4	sylanbrc	$⊢ F : A \underset{1-1}{⟶} B \to F : A \underset{1-1 onto}{⟶} ran F$