Metamath Proof Explorer

Theorem blssm

Description: A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Assertion blssm ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ( 𝑃 ( ball β€˜ 𝐷 ) 𝑅 ) βŠ† 𝑋 )

Proof

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