Definition df-crcts 27561

Description: Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)

Assertion

Ref	Expression
df-crcts	⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})

Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-crcts

Step	Hyp	Ref	Expression
1		ccrcts 27559	. 2 class Circuits
2		vg	. . 3 setvar 𝑔
3		cvv 3495	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1532	. . . . . 6 class 𝑓
6		vp	. . . . . . 7 setvar 𝑝
7	6	cv 1532	. . . . . 6 class 𝑝
8	2	cv 1532	. . . . . . 7 class 𝑔
9		ctrls 27466	. . . . . . 7 class Trails
10	8, 9	cfv 6350	. . . . . 6 class (Trails‘𝑔)
11	5, 7, 10	wbr 5059	. . . . 5 wff 𝑓(Trails‘𝑔)𝑝
12		cc0 10531	. . . . . . 7 class 0
13	12, 7	cfv 6350	. . . . . 6 class (𝑝‘0)
14		chash 13684	. . . . . . . 8 class ♯
15	5, 14	cfv 6350	. . . . . . 7 class (♯‘𝑓)
16	15, 7	cfv 6350	. . . . . 6 class (𝑝‘(♯‘𝑓))
17	13, 16	wceq 1533	. . . . 5 wff (𝑝‘0) = (𝑝‘(♯‘𝑓))
18	11, 17	wa 398	. . . 4 wff (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))
19	18, 4, 6	copab 5121	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
20	2, 3, 19	cmpt 5139	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
21	1, 20	wceq 1533	1 wff Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})

This definition is referenced by:  crcts  27563