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Definition df-clwlks 40972
Description: Define the set of all closed walks (in an undirected graph).

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 0 , ... , ( n - 1 ) } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clWlk 41293! (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.)

Assertion
Ref Expression
df-clwlks ClWalkS = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-clwlks
StepHypRef Expression
1 cclwlks 40971 . 2 class ClWalkS
2 vg . . 3 setvar 𝑔
3 cvv 3172 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1473 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1473 . . . . . 6 class 𝑝
82cv 1473 . . . . . . 7 class 𝑔
9 c1wlks 40791 . . . . . . 7 class 1Walks
108, 9cfv 5790 . . . . . 6 class (1Walks‘𝑔)
115, 7, 10wbr 4577 . . . . 5 wff 𝑓(1Walks‘𝑔)𝑝
12 cc0 9792 . . . . . . 7 class 0
1312, 7cfv 5790 . . . . . 6 class (𝑝‘0)
14 chash 12934 . . . . . . . 8 class #
155, 14cfv 5790 . . . . . . 7 class (#‘𝑓)
1615, 7cfv 5790 . . . . . 6 class (𝑝‘(#‘𝑓))
1713, 16wceq 1474 . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
1811, 17wa 382 . . . 4 wff (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
1918, 4, 6copab 4636 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
202, 3, 19cmpt 4637 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
211, 20wceq 1474 1 wff ClWalkS = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
Colors of variables: wff setvar class
This definition is referenced by:  clwlkS  40973
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