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Metamath Proof Explorer

Theorem f1ococnv2

Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003) (Proof shortened by Stefan O'Rear, 12-Feb-2015)

		Ref	Expression
	Assertion	f1ococnv2	$⊢ F : A \underset{1-1 onto}{⟶} B \to F \circ F^{-1} = {I ↾}_{B}$

Proof

Step	Hyp	Ref	Expression
1		f1ofo	$⊢ F : A \underset{1-1 onto}{⟶} B \to F : A \underset{onto}{⟶} B$
2		fococnv2	$⊢ F : A \underset{onto}{⟶} B \to F \circ F^{-1} = {I ↾}_{B}$
3	1 2	syl	$⊢ F : A \underset{1-1 onto}{⟶} B \to F \circ F^{-1} = {I ↾}_{B}$