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Definition df-line3 37703
Description: Define the set of all lines. A line is an infinite subset of RR3 that satisfies a PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.)
Assertion
Ref Expression
df-line3 line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2𝑜𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-line3
StepHypRef Expression
1 cline3 37687 . 2 class line3
2 c2o 7441 . . . . 5 class 2𝑜
3 vx . . . . . 6 setvar 𝑥
43cv 1474 . . . . 5 class 𝑥
5 cdom 7839 . . . . 5 class
62, 4, 5wbr 4583 . . . 4 wff 2𝑜𝑥
7 vz . . . . . . . . 9 setvar 𝑧
87cv 1474 . . . . . . . 8 class 𝑧
9 vy . . . . . . . . 9 setvar 𝑦
109cv 1474 . . . . . . . 8 class 𝑦
118, 10wne 2780 . . . . . . 7 wff 𝑧𝑦
1210, 8cptdfc 37685 . . . . . . . . 9 class PtDf(𝑦, 𝑧)
1312crn 5039 . . . . . . . 8 class ran PtDf(𝑦, 𝑧)
1413, 4wceq 1475 . . . . . . 7 wff ran PtDf(𝑦, 𝑧) = 𝑥
1511, 14wi 4 . . . . . 6 wff (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥)
1615, 7, 4wral 2896 . . . . 5 wff 𝑧𝑥 (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥)
1716, 9, 4wral 2896 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥)
186, 17wa 383 . . 3 wff (2𝑜𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))
19 crr3c 37686 . . . 4 class RR3
2019cpw 4108 . . 3 class 𝒫 RR3
2118, 3, 20crab 2900 . 2 class {𝑥 ∈ 𝒫 RR3 ∣ (2𝑜𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
221, 21wceq 1475 1 wff line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2𝑜𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑧𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
Colors of variables: wff setvar class
This definition is referenced by: (None)
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