Definition df-eupth 30142

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)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})

Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-eupth

Step	Hyp	Ref	Expression
1		ceupth 30141	. 2 class EulerPaths
2		vg	. . 3 setvar 𝑔
3		cvv 3436	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1539	. . . . . 6 class 𝑓
6		vp	. . . . . . 7 setvar 𝑝
7	6	cv 1539	. . . . . 6 class 𝑝
8	2	cv 1539	. . . . . . 7 class 𝑔
9		ctrls 29634	. . . . . . 7 class Trails
10	8, 9	cfv 6482	. . . . . 6 class (Trails‘𝑔)
11	5, 7, 10	wbr 5092	. . . . 5 wff 𝑓(Trails‘𝑔)𝑝
12		cc0 11009	. . . . . . 7 class 0
13		chash 14237	. . . . . . . 8 class ♯
14	5, 13	cfv 6482	. . . . . . 7 class (♯‘𝑓)
15		cfzo 13557	. . . . . . 7 class ..^
16	12, 14, 15	co 7349	. . . . . 6 class (0..^(♯‘𝑓))
17		ciedg 28942	. . . . . . . 8 class iEdg
18	8, 17	cfv 6482	. . . . . . 7 class (iEdg‘𝑔)
19	18	cdm 5619	. . . . . 6 class dom (iEdg‘𝑔)
20	16, 19, 5	wfo 6480	. . . . 5 wff 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔)
21	11, 20	wa 395	. . . 4 wff (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))
22	21, 4, 6	copab 5154	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))}
23	2, 3, 22	cmpt 5173	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
24	1, 23	wceq 1540	1 wff EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})

This definition is referenced by:  releupth  30143  eupths  30144