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Metamath Proof Explorer

Theorem mopnfss

Description: The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

		Ref	Expression
	Hypothesis	mopnval.1	$⊢ J = MetOpen (D)$
	Assertion	mopnfss	$⊢ D \in ∞Met (X) \to J \subseteq 𝒫 X$

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	$⊢ J = MetOpen (D)$
2		pwuni	$⊢ J \subseteq 𝒫 ⋃ J$
3	1	mopnuni	$⊢ D \in ∞Met (X) \to X = ⋃ J$
4	3	pweqd	$⊢ D \in ∞Met (X) \to 𝒫 X = 𝒫 ⋃ J$
5	2 4	sseqtrrid	$⊢ D \in ∞Met (X) \to J \subseteq 𝒫 X$