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Definition df-hlg 25214
Description: Define the function producting the relation "belong to the same half-line" (Contributed by Thierry Arnoux, 15-Aug-2020.)
Assertion
Ref Expression
df-hlg hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}))
Distinct variable group:   𝑎,𝑏,𝑐,𝑔

Detailed syntax breakdown of Definition df-hlg
StepHypRef Expression
1 chlg 25213 . 2 class hlG
2 vg . . 3 setvar 𝑔
3 cvv 3172 . . 3 class V
4 vc . . . 4 setvar 𝑐
52cv 1473 . . . . 5 class 𝑔
6 cbs 15641 . . . . 5 class Base
75, 6cfv 5790 . . . 4 class (Base‘𝑔)
8 va . . . . . . . . 9 setvar 𝑎
98cv 1473 . . . . . . . 8 class 𝑎
109, 7wcel 1976 . . . . . . 7 wff 𝑎 ∈ (Base‘𝑔)
11 vb . . . . . . . . 9 setvar 𝑏
1211cv 1473 . . . . . . . 8 class 𝑏
1312, 7wcel 1976 . . . . . . 7 wff 𝑏 ∈ (Base‘𝑔)
1410, 13wa 382 . . . . . 6 wff (𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔))
154cv 1473 . . . . . . . 8 class 𝑐
169, 15wne 2779 . . . . . . 7 wff 𝑎𝑐
1712, 15wne 2779 . . . . . . 7 wff 𝑏𝑐
18 citv 25052 . . . . . . . . . . 11 class Itv
195, 18cfv 5790 . . . . . . . . . 10 class (Itv‘𝑔)
2015, 12, 19co 6527 . . . . . . . . 9 class (𝑐(Itv‘𝑔)𝑏)
219, 20wcel 1976 . . . . . . . 8 wff 𝑎 ∈ (𝑐(Itv‘𝑔)𝑏)
2215, 9, 19co 6527 . . . . . . . . 9 class (𝑐(Itv‘𝑔)𝑎)
2312, 22wcel 1976 . . . . . . . 8 wff 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)
2421, 23wo 381 . . . . . . 7 wff (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))
2516, 17, 24w3a 1030 . . . . . 6 wff (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)))
2614, 25wa 382 . . . . 5 wff ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))
2726, 8, 11copab 4636 . . . 4 class {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}
284, 7, 27cmpt 4637 . . 3 class (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))})
292, 3, 28cmpt 4637 . 2 class (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}))
301, 29wceq 1474 1 wff hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}))
Colors of variables: wff setvar class
This definition is referenced by:  ishlg  25215
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