Theorem rntrclfv

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

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

Proof

Step	Hyp	Ref	Expression
1		trclfvdecoml	$$⊢R\in V\to \mathrm{t+}\left(R\right)=R\cup \left(R\circ \mathrm{t+}\left(R\right)\right)$$
2	1	rneqd	$$⊢R\in V\to \mathrm{ran}\mathrm{t+}\left(R\right)=\mathrm{ran}\left(R\cup \left(R\circ \mathrm{t+}\left(R\right)\right)\right)$$
3		rnun	$$⊢\mathrm{ran}\left(R\cup \left(R\circ \mathrm{t+}\left(R\right)\right)\right)=\mathrm{ran}R\cup \mathrm{ran}\left(R\circ \mathrm{t+}\left(R\right)\right)$$
4		rncoss	$$⊢\mathrm{ran}\left(R\circ \mathrm{t+}\left(R\right)\right)\subseteq \mathrm{ran}R$$
5		ssequn2	$$⊢\mathrm{ran}\left(R\circ \mathrm{t+}\left(R\right)\right)\subseteq \mathrm{ran}R\leftrightarrow \mathrm{ran}R\cup \mathrm{ran}\left(R\circ \mathrm{t+}\left(R\right)\right)=\mathrm{ran}R$$
6	4 5	mpbi	$$⊢\mathrm{ran}R\cup \mathrm{ran}\left(R\circ \mathrm{t+}\left(R\right)\right)=\mathrm{ran}R$$
7	3 6	eqtri	$$⊢\mathrm{ran}\left(R\cup \left(R\circ \mathrm{t+}\left(R\right)\right)\right)=\mathrm{ran}R$$
8	2 7	eqtrdi	$$⊢R\in V\to \mathrm{ran}\mathrm{t+}\left(R\right)=\mathrm{ran}R$$