Theorem brcnvin

Description: Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024)

	Ref	Expression
Assertion	brcnvin	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( 𝑅 ∩ ◡ 𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑆 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		brin	⊢ ( 𝐴 ( 𝑅 ∩ ◡ 𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐴 ◡ 𝑆 𝐵 ) )
2		brcnvg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ◡ 𝑆 𝐵 ↔ 𝐵 𝑆 𝐴 ) )
3	2	anbi2d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 𝑅 𝐵 ∧ 𝐴 ◡ 𝑆 𝐵 ) ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑆 𝐴 ) ) )
4	1 3	bitrid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( 𝑅 ∩ ◡ 𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑆 𝐴 ) ) )