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 = dom 𝐴

Proof

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