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