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Metamath Proof Explorer

Theorem rncnvcnv

Description: The range of the double converse of a class is equal to its range (even when that class in not a relation). (Contributed by NM, 8-Apr-2007)

		Ref	Expression
	Assertion	rncnvcnv	$⊢ ran {A^{-1}}^{-1} = ran A$

Proof

Step	Hyp	Ref	Expression
1		df-rn	$⊢ ran A = dom A^{-1}$
2		dfdm4	$⊢ dom A^{-1} = ran {A^{-1}}^{-1}$
3	1 2	eqtr2i	$⊢ ran {A^{-1}}^{-1} = ran A$