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Metamath Proof Explorer

Theorem iunconst

Description: Indexed union of a constant class, i.e. where B does not depend on x . (Contributed by NM, 5-Sep-2004) (Proof shortened by Andrew Salmon, 25-Jul-2011)

		Ref	Expression
	Assertion	iunconst	$⊢ A \neq \emptyset \to ⋃_{x \in A} B = B$

Proof

Step	Hyp	Ref	Expression
1		eliun	$⊢ y \in ⋃_{x \in A} B \leftrightarrow \exists x \in A y \in B$
2		r19.9rzv	$⊢ A \neq \emptyset \to (y \in B \leftrightarrow \exists x \in A y \in B)$
3	1 2	bitr4id	$⊢ A \neq \emptyset \to (y \in ⋃_{x \in A} B \leftrightarrow y \in B)$
4	3	eqrdv	$⊢ A \neq \emptyset \to ⋃_{x \in A} B = B$