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Definition df-cycl 26041
 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.)
Assertion
Ref Expression
df-cycl Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-cycl
StepHypRef Expression
1 ccycl 26035 . 2 class Cycles
2 vv . . 3 setvar 𝑣
3 ve . . 3 setvar 𝑒
4 cvv 3173 . . 3 class V
5 vf . . . . . . 7 setvar 𝑓
65cv 1474 . . . . . 6 class 𝑓
7 vp . . . . . . 7 setvar 𝑝
87cv 1474 . . . . . 6 class 𝑝
92cv 1474 . . . . . . 7 class 𝑣
103cv 1474 . . . . . . 7 class 𝑒
11 cpath 26028 . . . . . . 7 class Paths
129, 10, 11co 6549 . . . . . 6 class (𝑣 Paths 𝑒)
136, 8, 12wbr 4583 . . . . 5 wff 𝑓(𝑣 Paths 𝑒)𝑝
14 cc0 9815 . . . . . . 7 class 0
1514, 8cfv 5804 . . . . . 6 class (𝑝‘0)
16 chash 12979 . . . . . . . 8 class #
176, 16cfv 5804 . . . . . . 7 class (#‘𝑓)
1817, 8cfv 5804 . . . . . 6 class (𝑝‘(#‘𝑓))
1915, 18wceq 1475 . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
2013, 19wa 383 . . . 4 wff (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
2120, 5, 7copab 4642 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
222, 3, 4, 4, 21cmpt2 6551 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
231, 22wceq 1475 1 wff Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
 Colors of variables: wff setvar class This definition is referenced by:  cycls  26151  cyclispth  26157  cycliscrct  26158  cyclnspth  26159
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