Metamath Proof Explorer

Theorem opabresex2d

Description: Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017) (Revised by AV, 15-Jan-2021)

		Ref	Expression
	Hypotheses	opabresex2d.1	$⊢ (φ \land x W (G) y) \to ψ$
		opabresex2d.2	$⊢ φ \to \{⟨x, y⟩ \| ψ\} \in V$
	Assertion	opabresex2d	$⊢ φ \to \{⟨x, y⟩ \| (x W (G) y \land θ)\} \in V$

Proof

Step	Hyp	Ref	Expression
1		opabresex2d.1	$⊢ (φ \land x W (G) y) \to ψ$
2		opabresex2d.2	$⊢ φ \to \{⟨x, y⟩ \| ψ\} \in V$
3	1	ex	$⊢ φ \to (x W (G) y \to ψ)$
4	3	alrimivv	$⊢ φ \to \forall x \forall y (x W (G) y \to ψ)$
5		opabbrex	$⊢ (\forall x \forall y (x W (G) y \to ψ) \land \{⟨x, y⟩ \| ψ\} \in V) \to \{⟨x, y⟩ \| (x W (G) y \land θ)\} \in V$
6	4 2 5	syl2anc	$⊢ φ \to \{⟨x, y⟩ \| (x W (G) y \land θ)\} \in V$