Theorem brin

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

	Ref	Expression
Assertion	brin	$$⊢A\left(R\cap S\right)B\leftrightarrow \left(ARB\wedge ASB\right)$$

Proof

Step	Hyp	Ref	Expression
1		elin	$$⊢⟨A,B⟩\in \left(R\cap S\right)\leftrightarrow \left(⟨A,B⟩\in R\wedge ⟨A,B⟩\in S\right)$$
2		df-br	$$⊢A\left(R\cap S\right)B\leftrightarrow ⟨A,B⟩\in \left(R\cap S\right)$$
3		df-br	$$⊢ARB\leftrightarrow ⟨A,B⟩\in R$$
4		df-br	$$⊢ASB\leftrightarrow ⟨A,B⟩\in S$$
5	3 4	anbi12i	$$⊢\left(ARB\wedge ASB\right)\leftrightarrow \left(⟨A,B⟩\in R\wedge ⟨A,B⟩\in S\right)$$
6	1 2 5	3bitr4i	$$⊢A\left(R\cap S\right)B\leftrightarrow \left(ARB\wedge ASB\right)$$