Theorem mopnss

Description: An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006)

	Ref	Expression
Hypothesis	mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion	mopnss	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	mopntopon	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
3		toponss	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝑋 )
4	2 3	sylan	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝑋 )