Definition df-cycls 26682

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

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... 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-cycls	⊢ Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-cycls

Step	Hyp	Ref	Expression
1		ccycls 26680	. 2 class Cycles
2		vg	. . 3 setvar 𝑔
3		cvv 3200	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1482	. . . . . 6 class 𝑓
6		vp	. . . . . . 7 setvar 𝑝
7	6	cv 1482	. . . . . 6 class 𝑝
8	2	cv 1482	. . . . . . 7 class 𝑔
9		cpths 26608	. . . . . . 7 class Paths
10	8, 9	cfv 5888	. . . . . 6 class (Paths‘𝑔)
11	5, 7, 10	wbr 4653	. . . . 5 wff 𝑓(Paths‘𝑔)𝑝
12		cc0 9936	. . . . . . 7 class 0
13	12, 7	cfv 5888	. . . . . 6 class (𝑝‘0)
14		chash 13117	. . . . . . . 8 class #
15	5, 14	cfv 5888	. . . . . . 7 class (#‘𝑓)
16	15, 7	cfv 5888	. . . . . 6 class (𝑝‘(#‘𝑓))
17	13, 16	wceq 1483	. . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
18	11, 17	wa 384	. . . 4 wff (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
19	18, 4, 6	copab 4712	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
20	2, 3, 19	cmpt 4729	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
21	1, 20	wceq 1483	1 wff Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

This definition is referenced by:  cycls  26684