Theorem mopnin

Description: The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006) (Revised by Mario Carneiro, 23-Dec-2013)

	Ref	Expression
Hypothesis	mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion	mopnin	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	mopntop	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top )
3		inopn	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝐽 )
4	2 3	syl3an1	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝐽 )