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 ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		blf	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ball ‘ 𝐷 ) : ( 𝑋 × ℝ* ) ⟶ 𝒫 𝑋 )
2		fovrn	⊢ ( ( ( ball ‘ 𝐷 ) : ( 𝑋 × ℝ* ) ⟶ 𝒫 𝑋 ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ 𝒫 𝑋 )
3	1 2	syl3an1	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ 𝒫 𝑋 )
4	3	elpwid	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ⊆ 𝑋 )