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Definition df-wlks 40801
 Description: Define the set of all walks (in a pseudograph). TODO-AV: This corresponds to the definition of Walks, but can be removed and the defining theorem upgriswlk 40849 could be used instead. 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-wlks 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-wlks
StepHypRef Expression
1 cwlks 40797 . 2 class UPWalks
2 vg . . 3 setvar 𝑔
3 cvv 3173 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1474 . . . . . 6 class 𝑓
62cv 1474 . . . . . . . . 9 class 𝑔
7 ciedg 25674 . . . . . . . . 9 class iEdg
86, 7cfv 5804 . . . . . . . 8 class (iEdg‘𝑔)
98cdm 5038 . . . . . . 7 class dom (iEdg‘𝑔)
109cword 13146 . . . . . 6 class Word dom (iEdg‘𝑔)
115, 10wcel 1977 . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12 cc0 9815 . . . . . . 7 class 0
13 chash 12979 . . . . . . . 8 class #
145, 13cfv 5804 . . . . . . 7 class (#‘𝑓)
15 cfz 12197 . . . . . . 7 class ...
1612, 14, 15co 6549 . . . . . 6 class (0...(#‘𝑓))
17 cvtx 25673 . . . . . . 7 class Vtx
186, 17cfv 5804 . . . . . 6 class (Vtx‘𝑔)
19 vp . . . . . . 7 setvar 𝑝
2019cv 1474 . . . . . 6 class 𝑝
2116, 18, 20wf 5800 . . . . 5 wff 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔)
22 vk . . . . . . . . . 10 setvar 𝑘
2322cv 1474 . . . . . . . . 9 class 𝑘
2423, 5cfv 5804 . . . . . . . 8 class (𝑓𝑘)
2524, 8cfv 5804 . . . . . . 7 class ((iEdg‘𝑔)‘(𝑓𝑘))
2623, 20cfv 5804 . . . . . . . 8 class (𝑝𝑘)
27 c1 9816 . . . . . . . . . 10 class 1
28 caddc 9818 . . . . . . . . . 10 class +
2923, 27, 28co 6549 . . . . . . . . 9 class (𝑘 + 1)
3029, 20cfv 5804 . . . . . . . 8 class (𝑝‘(𝑘 + 1))
3126, 30cpr 4127 . . . . . . 7 class {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3225, 31wceq 1475 . . . . . 6 wff ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
33 cfzo 12334 . . . . . . 7 class ..^
3412, 14, 33co 6549 . . . . . 6 class (0..^(#‘𝑓))
3532, 22, 34wral 2896 . . . . 5 wff 𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3611, 21, 35w3a 1031 . . . 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 4642 . . 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 4643 . 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 1475 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:  wlksfval  40812
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