Theorem brcnvtrclfvcnv

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

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

Proof

Step	Hyp	Ref	Expression
1		cnvexg	$$⊢R\in U\to R^{-1}\in \mathrm{V}$$
2		brcnvtrclfv	$$⊢\left(R^{-1}\in \mathrm{V}\wedge A\in V\wedge B\in W\right)\to \left(A{\mathrm{t+}\left(R^{-1}\right)}^{-1}B\leftrightarrow \forall r\left(\left(R^{-1}\subseteq r\wedge r\circ r\subseteq r\right)\to BrA\right)\right)$$
3	1 2	syl3an1	$$⊢\left(R\in U\wedge A\in V\wedge B\in W\right)\to \left(A{\mathrm{t+}\left(R^{-1}\right)}^{-1}B\leftrightarrow \forall r\left(\left(R^{-1}\subseteq r\wedge r\circ r\subseteq r\right)\to BrA\right)\right)$$