Theorem releleccnv

Description: Elementhood in a converse R -coset when R is a relation. (Contributed by Peter Mazsa, 9-Dec-2018)

	Ref	Expression
Assertion	releleccnv	$$⊢\mathrm{Rel}R\to \left(A\in \left[B\right]R^{-1}\leftrightarrow ARB\right)$$

Proof

Step	Hyp	Ref	Expression
1		relcnv	$$⊢\mathrm{Rel}{R}^{-1}$$
2		relelec	$$⊢\mathrm{Rel}{R}^{-1}\to \left(A\in \left[B\right]R^{-1}\leftrightarrow B{R}^{-1}A\right)$$
3	1 2	ax-mp	$$⊢A\in \left[B\right]R^{-1}\leftrightarrow B{R}^{-1}A$$
4		relbrcnvg	$$⊢\mathrm{Rel}R\to \left(B{R}^{-1}A\leftrightarrow ARB\right)$$
5	3 4	syl5bb	$$⊢\mathrm{Rel}R\to \left(A\in \left[B\right]R^{-1}\leftrightarrow ARB\right)$$