Metamath Proof Explorer

Theorem opres

Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

		Ref	Expression
	Hypothesis	opres.1	$⊢ B \in V$
	Assertion	opres	$⊢ A \in D \to (⟨A, B⟩ \in ({C ↾}_{D}) \leftrightarrow ⟨A, B⟩ \in C)$

Proof

Step	Hyp	Ref	Expression
1		opres.1	$⊢ B \in V$
2	1	opelresi	$⊢ ⟨A, B⟩ \in ({C ↾}_{D}) \leftrightarrow (A \in D \land ⟨A, B⟩ \in C)$
3	2	baib	$⊢ A \in D \to (⟨A, B⟩ \in ({C ↾}_{D}) \leftrightarrow ⟨A, B⟩ \in C)$