Theorem brin

Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008)

	Ref	Expression
Assertion	brin	⊢ ( 𝐴 ( 𝑅 ∩ 𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐴 𝑆 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅 ∩ 𝑆 ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑆 ) )
2		df-br	⊢ ( 𝐴 ( 𝑅 ∩ 𝑆 ) 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅 ∩ 𝑆 ) )
3		df-br	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
4		df-br	⊢ ( 𝐴 𝑆 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑆 )
5	3 4	anbi12i	⊢ ( ( 𝐴 𝑅 𝐵 ∧ 𝐴 𝑆 𝐵 ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑆 ) )
6	1 2 5	3bitr4i	⊢ ( 𝐴 ( 𝑅 ∩ 𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐴 𝑆 𝐵 ) )