Metamath Proof Explorer

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 β€˜ 𝐷 )
Assertion mopnin ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝐡 ∈ 𝐽 ) β†’ ( 𝐴 ∩ 𝐡 ) ∈ 𝐽 )

Proof

Step Hyp Ref Expression
1 mopni.1 ⊒ 𝐽 = ( MetOpen β€˜ 𝐷 )
2 1 mopntop ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ 𝐽 ∈ Top )
3 inopn ⊒ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐡 ∈ 𝐽 ) β†’ ( 𝐴 ∩ 𝐡 ) ∈ 𝐽 )
4 2 3 syl3an1 ⊒ ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝐡 ∈ 𝐽 ) β†’ ( 𝐴 ∩ 𝐡 ) ∈ 𝐽 )