Database
BASIC TOPOLOGY
Metric spaces
Metric space balls
blelrnps
Metamath Proof Explorer
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 ) )