Metamath Proof Explorer


Theorem elmopn2

Description: A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Hypothesis mopnval.1 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion elmopn2 ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴𝐽 ↔ ( 𝐴𝑋 ∧ ∀ 𝑥𝐴𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 mopnval.1 𝐽 = ( MetOpen ‘ 𝐷 )
2 1 elmopn ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴𝐽 ↔ ( 𝐴𝑋 ∧ ∀ 𝑥𝐴𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥𝑧𝑧𝐴 ) ) ) )
3 ssel2 ( ( 𝐴𝑋𝑥𝐴 ) → 𝑥𝑋 )
4 blssex ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥𝑋 ) → ( ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥𝑧𝑧𝐴 ) ↔ ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
5 3 4 sylan2 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴𝑋𝑥𝐴 ) ) → ( ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥𝑧𝑧𝐴 ) ↔ ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
6 5 anassrs ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑥𝐴 ) → ( ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥𝑧𝑧𝐴 ) ↔ ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
7 6 ralbidva ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴𝑋 ) → ( ∀ 𝑥𝐴𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥𝑧𝑧𝐴 ) ↔ ∀ 𝑥𝐴𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
8 7 pm5.32da ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( 𝐴𝑋 ∧ ∀ 𝑥𝐴𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥𝑧𝑧𝐴 ) ) ↔ ( 𝐴𝑋 ∧ ∀ 𝑥𝐴𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) ) )
9 2 8 bitrd ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴𝐽 ↔ ( 𝐴𝑋 ∧ ∀ 𝑥𝐴𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) ) )