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Metamath Proof Explorer

Theorem iunin1

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in Enderton p. 30. Use uniiun to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015)

		Ref	Expression
	Assertion	iunin1	$⊢ ⋃_{x \in A} (C \cap B) = ⋃_{x \in A} C \cap B$

Proof

Step	Hyp	Ref	Expression
1		iunin2	$⊢ ⋃_{x \in A} (B \cap C) = B \cap ⋃_{x \in A} C$
2		incom	$⊢ C \cap B = B \cap C$
3	2	a1i	$⊢ x \in A \to C \cap B = B \cap C$
4	3	iuneq2i	$⊢ ⋃_{x \in A} (C \cap B) = ⋃_{x \in A} (B \cap C)$
5		incom	$⊢ ⋃_{x \in A} C \cap B = B \cap ⋃_{x \in A} C$
6	1 4 5	3eqtr4i	$⊢ ⋃_{x \in A} (C \cap B) = ⋃_{x \in A} C \cap B$