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Metamath Proof Explorer

Theorem opabssxp

Description: An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995)

		Ref	Expression
	Assertion	opabssxp	$⊢ \{⟨x, y⟩ \| ((x \in A \land y \in B) \land φ)\} \subseteq A \times B$

Step	Hyp	Ref	Expression
1		simpl	$⊢ ((x \in A \land y \in B) \land φ) \to (x \in A \land y \in B)$
2	1	ssopab2i	$⊢ \{⟨x, y⟩ \| ((x \in A \land y \in B) \land φ)\} \subseteq \{⟨x, y⟩ \| (x \in A \land y \in B)\}$
3		df-xp	$⊢ A \times B = \{⟨x, y⟩ \| (x \in A \land y \in B)\}$
4	2 3	sseqtrri	$⊢ \{⟨x, y⟩ \| ((x \in A \land y \in B) \land φ)\} \subseteq A \times B$