Metamath Proof Explorer

Theorem blssm

Description: A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Assertion blssm ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ⊆ X

Proof

Step Hyp Ref Expression
1 blf ⊢ D ∈ ∞Met ⁡ X → ball ⁡ D : X × ℝ * ⟶ 𝒫 X
2 fovrn ⊢ ball ⁡ D : X × ℝ * ⟶ 𝒫 X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ∈ 𝒫 X
3 1 2 syl3an1 ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ∈ 𝒫 X
4 3 elpwid ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ⊆ X