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Metamath Proof Explorer

Theorem xpexgALT

Description: Alternate proof of xpexg requiring Replacement ( ax-rep ) but not Power Set ( ax-pow ). (Contributed by Mario Carneiro, 20-May-2013) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	xpexgALT	$⊢ (A \in V \land B \in W) \to A \times B \in V$

Proof

Step	Hyp	Ref	Expression
1		iunid	$⊢ ⋃_{y \in B} \{y\} = B$
2	1	xpeq2i	$⊢ A \times ⋃_{y \in B} \{y\} = A \times B$
3		xpiundi	$⊢ A \times ⋃_{y \in B} \{y\} = ⋃_{y \in B} (A \times \{y\})$
4	2 3	eqtr3i	$⊢ A \times B = ⋃_{y \in B} (A \times \{y\})$
5		id	$⊢ B \in W \to B \in W$
6		fconstmpt	$⊢ A \times \{y\} = (x \in A ⟼ y)$
7		mptexg	$⊢ A \in V \to (x \in A ⟼ y) \in V$
8	6 7	eqeltrid	$⊢ A \in V \to A \times \{y\} \in V$
9	8	ralrimivw	$⊢ A \in V \to \forall y \in B A \times \{y\} \in V$
10		iunexg	$⊢ (B \in W \land \forall y \in B A \times \{y\} \in V) \to ⋃_{y \in B} (A \times \{y\}) \in V$
11	5 9 10	syl2anr	$⊢ (A \in V \land B \in W) \to ⋃_{y \in B} (A \times \{y\}) \in V$
12	4 11	eqeltrid	$⊢ (A \in V \land B \in W) \to A \times B \in V$