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Metamath Proof Explorer

Theorem f1cnv

Description: The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011)

		Ref	Expression
	Assertion	f1cnv	$⊢ F : A \underset{1-1}{⟶} B \to F^{-1} : ran F \underset{1-1 onto}{⟶} A$

Step	Hyp	Ref	Expression
1		f1f1orn	$⊢ F : A \underset{1-1}{⟶} B \to F : A \underset{1-1 onto}{⟶} ran F$
2		f1ocnv	$⊢ F : A \underset{1-1 onto}{⟶} ran F \to F^{-1} : ran F \underset{1-1 onto}{⟶} A$
3	1 2	syl	$⊢ F : A \underset{1-1}{⟶} B \to F^{-1} : ran F \underset{1-1 onto}{⟶} A$