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Definition df-upwlks 44349
 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
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 44348 . 2 class UPWalks
2 vg . . 3 setvar 𝑔
3 cvv 3444 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1537 . . . . . 6 class 𝑓
62cv 1537 . . . . . . . . 9 class 𝑔
7 ciedg 26793 . . . . . . . . 9 class iEdg
86, 7cfv 6328 . . . . . . . 8 class (iEdg‘𝑔)
98cdm 5523 . . . . . . 7 class dom (iEdg‘𝑔)
109cword 13861 . . . . . 6 class Word dom (iEdg‘𝑔)
115, 10wcel 2112 . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12 cc0 10530 . . . . . . 7 class 0
13 chash 13690 . . . . . . . 8 class
145, 13cfv 6328 . . . . . . 7 class (♯‘𝑓)
15 cfz 12889 . . . . . . 7 class ...
1612, 14, 15co 7139 . . . . . 6 class (0...(♯‘𝑓))
17 cvtx 26792 . . . . . . 7 class Vtx
186, 17cfv 6328 . . . . . 6 class (Vtx‘𝑔)
19 vp . . . . . . 7 setvar 𝑝
2019cv 1537 . . . . . 6 class 𝑝
2116, 18, 20wf 6324 . . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22 vk . . . . . . . . . 10 setvar 𝑘
2322cv 1537 . . . . . . . . 9 class 𝑘
2423, 5cfv 6328 . . . . . . . 8 class (𝑓𝑘)
2524, 8cfv 6328 . . . . . . 7 class ((iEdg‘𝑔)‘(𝑓𝑘))
2623, 20cfv 6328 . . . . . . . 8 class (𝑝𝑘)
27 c1 10531 . . . . . . . . . 10 class 1
28 caddc 10533 . . . . . . . . . 10 class +
2923, 27, 28co 7139 . . . . . . . . 9 class (𝑘 + 1)
3029, 20cfv 6328 . . . . . . . 8 class (𝑝‘(𝑘 + 1))
3126, 30cpr 4530 . . . . . . 7 class {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3225, 31wceq 1538 . . . . . 6 wff ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
33 cfzo 13032 . . . . . . 7 class ..^
3412, 14, 33co 7139 . . . . . 6 class (0..^(♯‘𝑓))
3532, 22, 34wral 3109 . . . . 5 wff 𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3611, 21, 35w3a 1084 . . . 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 5095 . . 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 5113 . 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 1538 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  44350
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