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Metamath Proof Explorer

Theorem blelrnps

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) (Revised by Thierry Arnoux, 11-Mar-2018)

Ref Expression
Assertion blelrnps ⊢ D ∈ PsMet ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ∈ ran ⁡ ball ⁡ D

Proof

Step Hyp Ref Expression
1 blfps ⊢ D ∈ PsMet ⁡ X → ball ⁡ D : X × ℝ * ⟶ 𝒫 X
2 1 ffnd ⊢ D ∈ PsMet ⁡ 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 ∈ PsMet ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R ∈ ran ⁡ ball ⁡ D