Theorem rncocnv2

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

	Ref	Expression
Assertion	rncocnv2	⊢ ran ( 𝐴 ∘ ◡ 𝐴 ) = ran 𝐴

Proof

Step	Hyp	Ref	Expression
1		dfdm4	⊢ dom 𝐴 = ran ◡ 𝐴
2		rncoeq	⊢ ( dom 𝐴 = ran ◡ 𝐴 → ran ( 𝐴 ∘ ◡ 𝐴 ) = ran 𝐴 )
3	1 2	ax-mp	⊢ ran ( 𝐴 ∘ ◡ 𝐴 ) = ran 𝐴