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	$$⊢\mathrm{dom}\left(A\circ A^{-1}\right)=\mathrm{ran}A$$

Proof

Step	Hyp	Ref	Expression
1		dfdm4	$$⊢\mathrm{dom}A=\mathrm{ran}{A}^{-1}$$
2		dmcoeq	$$⊢\mathrm{dom}A=\mathrm{ran}{A}^{-1}\to \mathrm{dom}\left(A\circ A^{-1}\right)=\mathrm{dom}{A}^{-1}$$
3	1 2	ax-mp	$$⊢\mathrm{dom}\left(A\circ A^{-1}\right)=\mathrm{dom}{A}^{-1}$$
4		df-rn	$$⊢\mathrm{ran}A=\mathrm{dom}{A}^{-1}$$
5	3 4	eqtr4i	$$⊢\mathrm{dom}\left(A\circ A^{-1}\right)=\mathrm{ran}A$$