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 β€˜ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ( 𝑃 ( ball β€˜ 𝐷 ) 𝑅 ) ∈ ran ( ball β€˜ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 blf ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ ( ball β€˜ 𝐷 ) : ( 𝑋 Γ— ℝ* ) ⟢ 𝒫 𝑋 )
2 1 ffnd ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ ( ball β€˜ 𝐷 ) Fn ( 𝑋 Γ— ℝ* ) )
3 fnovrn ⊒ ( ( ( ball β€˜ 𝐷 ) Fn ( 𝑋 Γ— ℝ* ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ( 𝑃 ( ball β€˜ 𝐷 ) 𝑅 ) ∈ ran ( ball β€˜ 𝐷 ) )
4 2 3 syl3an1 ⊒ ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ( 𝑃 ( ball β€˜ 𝐷 ) 𝑅 ) ∈ ran ( ball β€˜ 𝐷 ) )