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Metamath Proof Explorer

Theorem f1ocnvb

Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/range interchanged. (Contributed by NM, 8-Dec-2003)

		Ref	Expression
	Assertion	f1ocnvb	$⊢ Rel F \to (F : A \underset{1-1 onto}{⟶} B \leftrightarrow F^{-1} : B \underset{1-1 onto}{⟶} A)$

Proof

Step	Hyp	Ref	Expression
1		f1ocnv	$⊢ F : A \underset{1-1 onto}{⟶} B \to F^{-1} : B \underset{1-1 onto}{⟶} A$
2		f1ocnv	$⊢ F^{-1} : B \underset{1-1 onto}{⟶} A \to {F^{-1}}^{-1} : A \underset{1-1 onto}{⟶} B$
3		dfrel2	$⊢ Rel F \leftrightarrow {F^{-1}}^{-1} = F$
4		f1oeq1	$⊢ {F^{-1}}^{-1} = F \to ({F^{-1}}^{-1} : A \underset{1-1 onto}{⟶} B \leftrightarrow F : A \underset{1-1 onto}{⟶} B)$
5	3 4	sylbi	$⊢ Rel F \to ({F^{-1}}^{-1} : A \underset{1-1 onto}{⟶} B \leftrightarrow F : A \underset{1-1 onto}{⟶} B)$
6	2 5	imbitrid	$⊢ Rel F \to (F^{-1} : B \underset{1-1 onto}{⟶} A \to F : A \underset{1-1 onto}{⟶} B)$
7	1 6	impbid2	$⊢ Rel F \to (F : A \underset{1-1 onto}{⟶} B \leftrightarrow F^{-1} : B \underset{1-1 onto}{⟶} A)$