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Metamath Proof Explorer

Theorem blrnps

Description: Membership in the range of the ball function. Note that ran ( ballD ) is the collection of all balls for metric D . (Contributed by NM, 31-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013) (Revised by Thierry Arnoux, 11-Mar-2018)

Ref Expression
Assertion blrnps ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐴 ∈ ran ( ball ‘ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝑋 ∃ 𝑟 ∈ ℝ* 𝐴 = ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) )

Proof

Step Hyp Ref Expression
1 blfps ⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ball ‘ 𝐷 ) : ( 𝑋 × ℝ* ) ⟶ 𝒫 𝑋 )
2 ffn ⊢ ( ( ball ‘ 𝐷 ) : ( 𝑋 × ℝ* ) ⟶ 𝒫 𝑋 → ( ball ‘ 𝐷 ) Fn ( 𝑋 × ℝ* ) )
3 ovelrn ⊢ ( ( ball ‘ 𝐷 ) Fn ( 𝑋 × ℝ* ) → ( 𝐴 ∈ ran ( ball ‘ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝑋 ∃ 𝑟 ∈ ℝ* 𝐴 = ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) )
4 1 2 3 3syl ⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐴 ∈ ran ( ball ‘ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝑋 ∃ 𝑟 ∈ ℝ* 𝐴 = ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) )