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Definition df-clwlk 26040
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 { 1 , ... , n } 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(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 26055! (Contributed by Alexander van der Vekens, 12-Mar-2018.)

Assertion
Ref Expression
df-clwlk ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-clwlk
StepHypRef Expression
1 cclwlk 26037 . 2 class ClWalks
2 vv . . 3 setvar 𝑣
3 ve . . 3 setvar 𝑒
4 cvv 3168 . . 3 class V
5 vf . . . . . . 7 setvar 𝑓
65cv 1473 . . . . . 6 class 𝑓
7 vp . . . . . . 7 setvar 𝑝
87cv 1473 . . . . . 6 class 𝑝
92cv 1473 . . . . . . 7 class 𝑣
103cv 1473 . . . . . . 7 class 𝑒
11 cwalk 25788 . . . . . . 7 class Walks
129, 10, 11co 6523 . . . . . 6 class (𝑣 Walks 𝑒)
136, 8, 12wbr 4573 . . . . 5 wff 𝑓(𝑣 Walks 𝑒)𝑝
14 cc0 9788 . . . . . . 7 class 0
1514, 8cfv 5786 . . . . . 6 class (𝑝‘0)
16 chash 12930 . . . . . . . 8 class #
176, 16cfv 5786 . . . . . . 7 class (#‘𝑓)
1817, 8cfv 5786 . . . . . 6 class (𝑝‘(#‘𝑓))
1915, 18wceq 1474 . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
2013, 19wa 382 . . . 4 wff (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
2120, 5, 7copab 4632 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
222, 3, 4, 4, 21cmpt2 6525 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
231, 22wceq 1474 1 wff ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
Colors of variables: wff setvar class
This definition is referenced by:  clwlk  26043  isclwlkg  26045  clwlkiswlk  26047  clwlkswlks  26048  clwlkcompim  26054
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