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<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 pseudograhs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)

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)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})

Distinct variable group:   𝑓,𝑔,𝑘,𝑝

Detailed syntax breakdown of Definition df-wlks

Step	Hyp	Ref	Expression
1		cwlks 40797	. 2 class UPWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3173	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1474	. . . . . 6 class 𝑓
6	2	cv 1474	. . . . . . . . 9 class 𝑔
7		ciedg 25674	. . . . . . . . 9 class iEdg
8	6, 7	cfv 5804	. . . . . . . 8 class (iEdg‘𝑔)
9	8	cdm 5038	. . . . . . 7 class dom (iEdg‘𝑔)
10	9	cword 13146	. . . . . 6 class Word dom (iEdg‘𝑔)
11	5, 10	wcel 1977	. . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12		cc0 9815	. . . . . . 7 class 0
13		chash 12979	. . . . . . . 8 class #
14	5, 13	cfv 5804	. . . . . . 7 class (#‘𝑓)
15		cfz 12197	. . . . . . 7 class ...
16	12, 14, 15	co 6549	. . . . . 6 class (0...(#‘𝑓))
17		cvtx 25673	. . . . . . 7 class Vtx
18	6, 17	cfv 5804	. . . . . 6 class (Vtx‘𝑔)
19		vp	. . . . . . 7 setvar 𝑝
20	19	cv 1474	. . . . . 6 class 𝑝
21	16, 18, 20	wf 5800	. . . . 5 wff 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔)
22		vk	. . . . . . . . . 10 setvar 𝑘
23	22	cv 1474	. . . . . . . . 9 class 𝑘
24	23, 5	cfv 5804	. . . . . . . 8 class (𝑓‘𝑘)
25	24, 8	cfv 5804	. . . . . . 7 class ((iEdg‘𝑔)‘(𝑓‘𝑘))
26	23, 20	cfv 5804	. . . . . . . 8 class (𝑝‘𝑘)
27		c1 9816	. . . . . . . . . 10 class 1
28		caddc 9818	. . . . . . . . . 10 class +
29	23, 27, 28	co 6549	. . . . . . . . 9 class (𝑘 + 1)
30	29, 20	cfv 5804	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
31	26, 30	cpr 4127	. . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
32	25, 31	wceq 1475	. . . . . 6 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
33		cfzo 12334	. . . . . . 7 class ..^
34	12, 14, 33	co 6549	. . . . . 6 class (0..^(#‘𝑓))
35	32, 22, 34	wral 2896	. . . . 5 wff ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
36	11, 21, 35	w3a 1031	. . . 4 wff (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})
37	36, 4, 19	copab 4642	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}
38	2, 3, 37	cmpt 4643	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
39	1, 38	wceq 1475	1 wff UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})

This definition is referenced by:  wlksfval  40812