Theorem dmcocnv2

Description: The domain of a class precomposed with its converse is equal to its range. (Contributed by BJ, 25-Oct-2026)

	Ref	Expression
Assertion	dmcocnv2	⊢ dom ( 𝐴 ∘ ◡ 𝐴 ) = ran 𝐴

Proof

Step	Hyp	Ref	Expression
1		dfdm4	⊢ dom 𝐴 = ran ◡ 𝐴
2		dmcoeq	⊢ ( dom 𝐴 = ran ◡ 𝐴 → dom ( 𝐴 ∘ ◡ 𝐴 ) = dom ◡ 𝐴 )
3	1 2	ax-mp	⊢ dom ( 𝐴 ∘ ◡ 𝐴 ) = dom ◡ 𝐴
4		df-rn	⊢ ran 𝐴 = dom ◡ 𝐴
5	3 4	eqtr4i	⊢ dom ( 𝐴 ∘ ◡ 𝐴 ) = ran 𝐴