Theorem dmtrclfvRP

Description: The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	dmtrclfvRP	$$⊢R\in V\to \mathrm{dom}\mathrm{t+}\left(R\right)=\mathrm{dom}R$$

Proof

Step	Hyp	Ref	Expression
1		trclfvdecomr	$$⊢R\in V\to \mathrm{t+}\left(R\right)=R\cup \left(\mathrm{t+}\left(R\right)\circ R\right)$$
2	1	dmeqd	$$⊢R\in V\to \mathrm{dom}\mathrm{t+}\left(R\right)=\mathrm{dom}\left(R\cup \left(\mathrm{t+}\left(R\right)\circ R\right)\right)$$
3		dmun	$$⊢\mathrm{dom}\left(R\cup \left(\mathrm{t+}\left(R\right)\circ R\right)\right)=\mathrm{dom}R\cup \mathrm{dom}\left(\mathrm{t+}\left(R\right)\circ R\right)$$
4		dmcoss	$$⊢\mathrm{dom}\left(\mathrm{t+}\left(R\right)\circ R\right)\subseteq \mathrm{dom}R$$
5		ssequn2	$$⊢\mathrm{dom}\left(\mathrm{t+}\left(R\right)\circ R\right)\subseteq \mathrm{dom}R\leftrightarrow \mathrm{dom}R\cup \mathrm{dom}\left(\mathrm{t+}\left(R\right)\circ R\right)=\mathrm{dom}R$$
6	4 5	mpbi	$$⊢\mathrm{dom}R\cup \mathrm{dom}\left(\mathrm{t+}\left(R\right)\circ R\right)=\mathrm{dom}R$$
7	3 6	eqtri	$$⊢\mathrm{dom}\left(R\cup \left(\mathrm{t+}\left(R\right)\circ R\right)\right)=\mathrm{dom}R$$
8	2 7	eqtrdi	$$⊢R\in V\to \mathrm{dom}\mathrm{t+}\left(R\right)=\mathrm{dom}R$$