Theorem brin2

Description: Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021)

	Ref	Expression
Assertion	brin2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( 𝑅 ∩ 𝑆 ) 𝐵 ↔ 𝐴 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝐵 , 𝐵 ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		brin	⊢ ( 𝐴 ( 𝑅 ∩ 𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐴 𝑆 𝐵 ) )
2		brxrn	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝐵 , 𝐵 ⟩ ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐴 𝑆 𝐵 ) ) )
3	2	3anidm23	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝐵 , 𝐵 ⟩ ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐴 𝑆 𝐵 ) ) )
4	1 3	bitr4id	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( 𝑅 ∩ 𝑆 ) 𝐵 ↔ 𝐴 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝐵 , 𝐵 ⟩ ) )