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 ) ∧ 𝑃𝑋𝑅 ∈ 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* ) ∧ 𝑃𝑋𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ ran ( ball ‘ 𝐷 ) )
4 2 3 syl3an1 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃𝑋𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ ran ( ball ‘ 𝐷 ) )