Theorem brtrclfvcnv

Description: Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020)

	Ref	Expression
Assertion	brtrclfvcnv	$$⊢R\in V\to \left(A\mathrm{t+}\left(R^{-1}\right)B\leftrightarrow \forall r\left(\left(R^{-1}\subseteq r\wedge r\circ r\subseteq r\right)\to ArB\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cnvexg	$$⊢R\in V\to R^{-1}\in \mathrm{V}$$
2		brtrclfv	$$⊢R^{-1}\in \mathrm{V}\to \left(A\mathrm{t+}\left(R^{-1}\right)B\leftrightarrow \forall r\left(\left(R^{-1}\subseteq r\wedge r\circ r\subseteq r\right)\to ArB\right)\right)$$
3	1 2	syl	$$⊢R\in V\to \left(A\mathrm{t+}\left(R^{-1}\right)B\leftrightarrow \forall r\left(\left(R^{-1}\subseteq r\wedge r\circ r\subseteq r\right)\to ArB\right)\right)$$