Definition df-upwlks 48625

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)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})

Distinct variable group:   𝑓,𝑔,𝑘,𝑝

Detailed syntax breakdown of Definition df-upwlks

Step	Hyp	Ref	Expression
1		cupwlks 48624	. 2 class UPWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3430	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1541	. . . . . 6 class 𝑓
6	2	cv 1541	. . . . . . . . 9 class 𝑔
7		ciedg 29083	. . . . . . . . 9 class iEdg
8	6, 7	cfv 6493	. . . . . . . 8 class (iEdg‘𝑔)
9	8	cdm 5625	. . . . . . 7 class dom (iEdg‘𝑔)
10	9	cword 14469	. . . . . 6 class Word dom (iEdg‘𝑔)
11	5, 10	wcel 2114	. . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12		cc0 11032	. . . . . . 7 class 0
13		chash 14286	. . . . . . . 8 class ♯
14	5, 13	cfv 6493	. . . . . . 7 class (♯‘𝑓)
15		cfz 13455	. . . . . . 7 class ...
16	12, 14, 15	co 7361	. . . . . 6 class (0...(♯‘𝑓))
17		cvtx 29082	. . . . . . 7 class Vtx
18	6, 17	cfv 6493	. . . . . 6 class (Vtx‘𝑔)
19		vp	. . . . . . 7 setvar 𝑝
20	19	cv 1541	. . . . . 6 class 𝑝
21	16, 18, 20	wf 6489	. . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22		vk	. . . . . . . . . 10 setvar 𝑘
23	22	cv 1541	. . . . . . . . 9 class 𝑘
24	23, 5	cfv 6493	. . . . . . . 8 class (𝑓‘𝑘)
25	24, 8	cfv 6493	. . . . . . 7 class ((iEdg‘𝑔)‘(𝑓‘𝑘))
26	23, 20	cfv 6493	. . . . . . . 8 class (𝑝‘𝑘)
27		c1 11033	. . . . . . . . . 10 class 1
28		caddc 11035	. . . . . . . . . 10 class +
29	23, 27, 28	co 7361	. . . . . . . . 9 class (𝑘 + 1)
30	29, 20	cfv 6493	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
31	26, 30	cpr 4570	. . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
32	25, 31	wceq 1542	. . . . . 6 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
33		cfzo 13602	. . . . . . 7 class ..^
34	12, 14, 33	co 7361	. . . . . 6 class (0..^(♯‘𝑓))
35	32, 22, 34	wral 3052	. . . . 5 wff ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
36	11, 21, 35	w3a 1087	. . . 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 5148	. . 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 5167	. 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 1542	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:  upwlksfval  48626