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)
Ref | Expression | ||
---|---|---|---|
Assertion | blrn | ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ ran ( ball ‘ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝑋 ∃ 𝑟 ∈ ℝ* 𝐴 = ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blf | ⊢ ( 𝐷 ∈ ( ∞Met ‘ 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 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ ran ( ball ‘ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝑋 ∃ 𝑟 ∈ ℝ* 𝐴 = ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |