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Metamath Proof Explorer

Theorem dmcnvcnv

Description: The domain of the double converse of a class is equal to its domain (even when that class in not a relation, in which case dfrel2 gives another proof). (Contributed by NM, 8-Apr-2007)

		Ref	Expression
	Assertion	dmcnvcnv	$⊢ dom {A^{-1}}^{-1} = dom A$

Proof

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