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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 ) → ( 𝐴 ∈ 𝐽 ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) ) )

Proof

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