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Metamath Proof Explorer

Theorem 2iunin

Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008)

		Ref	Expression
	Assertion	2iunin	$⊢ ⋃_{x \in A} ⋃_{y \in B} (C \cap D) = ⋃_{x \in A} C \cap ⋃_{y \in B} D$

Step	Hyp	Ref	Expression
1		iunin2	$⊢ ⋃_{y \in B} (C \cap D) = C \cap ⋃_{y \in B} D$
2	1	a1i	$⊢ x \in A \to ⋃_{y \in B} (C \cap D) = C \cap ⋃_{y \in B} D$
3	2	iuneq2i	$⊢ ⋃_{x \in A} ⋃_{y \in B} (C \cap D) = ⋃_{x \in A} (C \cap ⋃_{y \in B} D)$
4		iunin1	$⊢ ⋃_{x \in A} (C \cap ⋃_{y \in B} D) = ⋃_{x \in A} C \cap ⋃_{y \in B} D$
5	3 4	eqtri	$⊢ ⋃_{x \in A} ⋃_{y \in B} (C \cap D) = ⋃_{x \in A} C \cap ⋃_{y \in B} D$