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Metamath Proof Explorer

Theorem xpv

Description: Cartesian product of a class and the universe. (Contributed by Peter Mazsa, 6-Oct-2020)

		Ref	Expression
	Assertion	xpv	$⊢ A \times V = \{⟨x, y⟩ \| x \in A\}$

Step	Hyp	Ref	Expression
1		df-xp	$⊢ A \times V = \{⟨x, y⟩ \| (x \in A \land y \in V)\}$
2		vex	$⊢ y \in V$
3		iba	$⊢ y \in V \to (x \in A \leftrightarrow (x \in A \land y \in V))$
4	2 3	ax-mp	$⊢ x \in A \leftrightarrow (x \in A \land y \in V)$
5	4	opabbii	$⊢ \{⟨x, y⟩ \| x \in A\} = \{⟨x, y⟩ \| (x \in A \land y \in V)\}$
6	1 5	eqtr4i	$⊢ A \times V = \{⟨x, y⟩ \| x \in A\}$