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Metamath Proof Explorer

Theorem mopni

Description: An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Hypothesis mopni.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion mopni ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 mopni.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2 1 elmopn ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ 𝐽 ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) ) )
3 2 simplbda ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )
4 eleq1 ⊢ ( 𝑦 = 𝑃 → ( 𝑦 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥 ) )
5 4 anbi1d ⊢ ( 𝑦 = 𝑃 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ↔ ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
6 5 rexbidv ⊢ ( 𝑦 = 𝑃 → ( ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
7 6 rspccv ⊢ ( ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) → ( 𝑃 ∈ 𝐴 → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
8 3 7 syl ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → ( 𝑃 ∈ 𝐴 → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
9 8 3impia ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )