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Definition df-upwlks 45937
Description: Define the set of all walks (in a pseudograph), called "simple walks" in the following.

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the 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).

Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)

Assertion
Ref Expression
df-upwlks UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
Distinct variable group:   𝑓,𝑔,𝑘,𝑝

Detailed syntax breakdown of Definition df-upwlks
StepHypRef Expression
1 cupwlks 45936 . 2 class UPWalks
2 vg . . 3 setvar 𝑔
3 cvv 3443 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1540 . . . . . 6 class 𝑓
62cv 1540 . . . . . . . . 9 class 𝑔
7 ciedg 27777 . . . . . . . . 9 class iEdg
86, 7cfv 6493 . . . . . . . 8 class (iEdg‘𝑔)
98cdm 5631 . . . . . . 7 class dom (iEdg‘𝑔)
109cword 14356 . . . . . 6 class Word dom (iEdg‘𝑔)
115, 10wcel 2106 . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12 cc0 11009 . . . . . . 7 class 0
13 chash 14184 . . . . . . . 8 class
145, 13cfv 6493 . . . . . . 7 class (♯‘𝑓)
15 cfz 13378 . . . . . . 7 class ...
1612, 14, 15co 7351 . . . . . 6 class (0...(♯‘𝑓))
17 cvtx 27776 . . . . . . 7 class Vtx
186, 17cfv 6493 . . . . . 6 class (Vtx‘𝑔)
19 vp . . . . . . 7 setvar 𝑝
2019cv 1540 . . . . . 6 class 𝑝
2116, 18, 20wf 6489 . . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22 vk . . . . . . . . . 10 setvar 𝑘
2322cv 1540 . . . . . . . . 9 class 𝑘
2423, 5cfv 6493 . . . . . . . 8 class (𝑓𝑘)
2524, 8cfv 6493 . . . . . . 7 class ((iEdg‘𝑔)‘(𝑓𝑘))
2623, 20cfv 6493 . . . . . . . 8 class (𝑝𝑘)
27 c1 11010 . . . . . . . . . 10 class 1
28 caddc 11012 . . . . . . . . . 10 class +
2923, 27, 28co 7351 . . . . . . . . 9 class (𝑘 + 1)
3029, 20cfv 6493 . . . . . . . 8 class (𝑝‘(𝑘 + 1))
3126, 30cpr 4586 . . . . . . 7 class {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3225, 31wceq 1541 . . . . . 6 wff ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
33 cfzo 13521 . . . . . . 7 class ..^
3412, 14, 33co 7351 . . . . . 6 class (0..^(♯‘𝑓))
3532, 22, 34wral 3062 . . . . 5 wff 𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3611, 21, 35w3a 1087 . . . 4 wff (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})
3736, 4, 19copab 5165 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})}
382, 3, 37cmpt 5186 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
391, 38wceq 1541 1 wff UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
Colors of variables: wff setvar class
This definition is referenced by:  upwlksfval  45938
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