Theorem dmcoss3

Description: The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019)

	Ref	Expression
Assertion	dmcoss3	$$⊢\mathrm{dom}\wr R=\mathrm{dom}{R}^{-1}$$

Proof

Step	Hyp	Ref	Expression
1		dfcoss3	$$⊢\wr R=R\circ R^{-1}$$
2	1	dmeqi	$$⊢\mathrm{dom}\wr R=\mathrm{dom}\left(R\circ R^{-1}\right)$$
3		rncnv	$$⊢\mathrm{ran}{R}^{-1}=\mathrm{dom}R$$
4	3	eqimssi	$$⊢\mathrm{ran}{R}^{-1}\subseteq \mathrm{dom}R$$
5		dmcosseq	$$⊢\mathrm{ran}{R}^{-1}\subseteq \mathrm{dom}R\to \mathrm{dom}\left(R\circ R^{-1}\right)=\mathrm{dom}{R}^{-1}$$
6	4 5	ax-mp	$$⊢\mathrm{dom}\left(R\circ R^{-1}\right)=\mathrm{dom}{R}^{-1}$$
7	2 6	eqtri	$$⊢\mathrm{dom}\wr R=\mathrm{dom}{R}^{-1}$$