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Metamath Proof Explorer

Theorem brin3

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

		Ref	Expression
	Assertion	brin3	$⊢ (A \in V \land B \in W) \to (A (R \cap S) B \leftrightarrow A R ⋉ S \{\{B\}\})$

Proof

Step	Hyp	Ref	Expression
1		brin2	$⊢ (A \in V \land B \in W) \to (A (R \cap S) B \leftrightarrow A R ⋉ S ⟨B, B⟩)$
2		opidg	$⊢ B \in W \to ⟨B, B⟩ = \{\{B\}\}$
3	2	adantl	$⊢ (A \in V \land B \in W) \to ⟨B, B⟩ = \{\{B\}\}$
4	3	breq2d	$⊢ (A \in V \land B \in W) \to (A R ⋉ S ⟨B, B⟩ \leftrightarrow A R ⋉ S \{\{B\}\})$
5	1 4	bitrd	$⊢ (A \in V \land B \in W) \to (A (R \cap S) B \leftrightarrow A R ⋉ S \{\{B\}\})$