Metamath Proof Explorer

Theorem blelrn

Description: A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

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

Proof

Step Hyp Ref Expression
1 blf ⊢ D ∈ ∞Met ⁡ X → ball ⁡ D : X × ℝ * ⟶ 𝒫 X
2 1 ffnd ⊢ D ∈ ∞Met ⁡ X → ball ⁡ D Fn X × ℝ *
3 fnovrn ⊢ ball ⁡ D Fn X × ℝ * ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ∈ ran ⁡ ball ⁡ D
4 2 3 syl3an1 ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ∈ ran ⁡ ball ⁡ D