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Definition df-eupth 27904
Description: Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
Assertion
Ref Expression
df-eupth EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-eupth
StepHypRef Expression
1 ceupth 27903 . 2 class EulerPaths
2 vg . . 3 setvar 𝑔
3 cvv 3492 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1527 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1527 . . . . . 6 class 𝑝
82cv 1527 . . . . . . 7 class 𝑔
9 ctrls 27399 . . . . . . 7 class Trails
108, 9cfv 6348 . . . . . 6 class (Trails‘𝑔)
115, 7, 10wbr 5057 . . . . 5 wff 𝑓(Trails‘𝑔)𝑝
12 cc0 10525 . . . . . . 7 class 0
13 chash 13678 . . . . . . . 8 class
145, 13cfv 6348 . . . . . . 7 class (♯‘𝑓)
15 cfzo 13021 . . . . . . 7 class ..^
1612, 14, 15co 7145 . . . . . 6 class (0..^(♯‘𝑓))
17 ciedg 26709 . . . . . . . 8 class iEdg
188, 17cfv 6348 . . . . . . 7 class (iEdg‘𝑔)
1918cdm 5548 . . . . . 6 class dom (iEdg‘𝑔)
2016, 19, 5wfo 6346 . . . . 5 wff 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔)
2111, 20wa 396 . . . 4 wff (𝑓(Trails‘𝑔)𝑝𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))
2221, 4, 6copab 5119 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))}
232, 3, 22cmpt 5137 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
241, 23wceq 1528 1 wff EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
Colors of variables: wff setvar class
This definition is referenced by:  releupth  27905  eupths  27906
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