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Definition df-upwlks 48050
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 48049 . 2 class UPWalks
2 vg . . 3 setvar 𝑔
3 cvv 3480 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1539 . . . . . 6 class 𝑓
62cv 1539 . . . . . . . . 9 class 𝑔
7 ciedg 29014 . . . . . . . . 9 class iEdg
86, 7cfv 6561 . . . . . . . 8 class (iEdg‘𝑔)
98cdm 5685 . . . . . . 7 class dom (iEdg‘𝑔)
109cword 14552 . . . . . 6 class Word dom (iEdg‘𝑔)
115, 10wcel 2108 . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12 cc0 11155 . . . . . . 7 class 0
13 chash 14369 . . . . . . . 8 class
145, 13cfv 6561 . . . . . . 7 class (♯‘𝑓)
15 cfz 13547 . . . . . . 7 class ...
1612, 14, 15co 7431 . . . . . 6 class (0...(♯‘𝑓))
17 cvtx 29013 . . . . . . 7 class Vtx
186, 17cfv 6561 . . . . . 6 class (Vtx‘𝑔)
19 vp . . . . . . 7 setvar 𝑝
2019cv 1539 . . . . . 6 class 𝑝
2116, 18, 20wf 6557 . . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22 vk . . . . . . . . . 10 setvar 𝑘
2322cv 1539 . . . . . . . . 9 class 𝑘
2423, 5cfv 6561 . . . . . . . 8 class (𝑓𝑘)
2524, 8cfv 6561 . . . . . . 7 class ((iEdg‘𝑔)‘(𝑓𝑘))
2623, 20cfv 6561 . . . . . . . 8 class (𝑝𝑘)
27 c1 11156 . . . . . . . . . 10 class 1
28 caddc 11158 . . . . . . . . . 10 class +
2923, 27, 28co 7431 . . . . . . . . 9 class (𝑘 + 1)
3029, 20cfv 6561 . . . . . . . 8 class (𝑝‘(𝑘 + 1))
3126, 30cpr 4628 . . . . . . 7 class {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3225, 31wceq 1540 . . . . . 6 wff ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
33 cfzo 13694 . . . . . . 7 class ..^
3412, 14, 33co 7431 . . . . . 6 class (0..^(♯‘𝑓))
3532, 22, 34wral 3061 . . . . 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 5205 . . 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 5225 . 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 1540 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  48051
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