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Definition df-bl 19510
Description: Define the metric space ball function. See blval 21948 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
df-bl ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
Distinct variable group:   𝑥,𝑑,𝑦,𝑧

Detailed syntax breakdown of Definition df-bl
StepHypRef Expression
1 cbl 19502 . 2 class ball
2 vd . . 3 setvar 𝑑
3 cvv 3172 . . 3 class V
4 vx . . . 4 setvar 𝑥
5 vz . . . 4 setvar 𝑧
62cv 1473 . . . . . 6 class 𝑑
76cdm 5027 . . . . 5 class dom 𝑑
87cdm 5027 . . . 4 class dom dom 𝑑
9 cxr 9929 . . . 4 class *
104cv 1473 . . . . . . 7 class 𝑥
11 vy . . . . . . . 8 setvar 𝑦
1211cv 1473 . . . . . . 7 class 𝑦
1310, 12, 6co 6526 . . . . . 6 class (𝑥𝑑𝑦)
145cv 1473 . . . . . 6 class 𝑧
15 clt 9930 . . . . . 6 class <
1613, 14, 15wbr 4577 . . . . 5 wff (𝑥𝑑𝑦) < 𝑧
1716, 11, 8crab 2899 . . . 4 class {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}
184, 5, 8, 9, 17cmpt2 6528 . . 3 class (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧})
192, 3, 18cmpt 4637 . 2 class (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
201, 19wceq 1474 1 wff ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
Colors of variables: wff setvar class
This definition is referenced by:  blfvalps  21945
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