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	⊢ ( Rel 𝑅 → ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐴 𝑅 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		relcnv	⊢ Rel ◡ 𝑅
2		relelec	⊢ ( Rel ◡ 𝑅 → ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐵 ◡ 𝑅 𝐴 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐵 ◡ 𝑅 𝐴 )
4		relbrcnvg	⊢ ( Rel 𝑅 → ( 𝐵 ◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝐵 ) )
5	3 4	bitrid	⊢ ( Rel 𝑅 → ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐴 𝑅 𝐵 ) )