Definition df-upwlks 48977

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 48976	. 2 class UPWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3457	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1569	. . . . . 6 class 𝑓
6	2	cv 1569	. . . . . . . . 9 class 𝑔
7		ciedg 29407	. . . . . . . . 9 class iEdg
8	6, 7	cfv 6541	. . . . . . . 8 class (iEdg‘𝑔)
9	8	cdm 5663	. . . . . . 7 class dom (iEdg‘𝑔)
10	9	cword 14573	. . . . . 6 class Word dom (iEdg‘𝑔)
11	5, 10	wcel 2146	. . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12		cc0 11120	. . . . . . 7 class 0
13		chash 14389	. . . . . . . 8 class ♯
14	5, 13	cfv 6541	. . . . . . 7 class (♯‘𝑓)
15		cfz 13556	. . . . . . 7 class ...
16	12, 14, 15	co 7420	. . . . . 6 class (0...(♯‘𝑓))
17		cvtx 29406	. . . . . . 7 class Vtx
18	6, 17	cfv 6541	. . . . . 6 class (Vtx‘𝑔)
19		vp	. . . . . . 7 setvar 𝑝
20	19	cv 1569	. . . . . 6 class 𝑝
21	16, 18, 20	wf 6537	. . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22		vk	. . . . . . . . . 10 setvar 𝑘
23	22	cv 1569	. . . . . . . . 9 class 𝑘
24	23, 5	cfv 6541	. . . . . . . 8 class (𝑓‘𝑘)
25	24, 8	cfv 6541	. . . . . . 7 class ((iEdg‘𝑔)‘(𝑓‘𝑘))
26	23, 20	cfv 6541	. . . . . . . 8 class (𝑝‘𝑘)
27		c1 11121	. . . . . . . . . 10 class 1
28		caddc 11123	. . . . . . . . . 10 class +
29	23, 27, 28	co 7420	. . . . . . . . 9 class (𝑘 + 1)
30	29, 20	cfv 6541	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
31	26, 30	cpr 4593	. . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
32	25, 31	wceq 1570	. . . . . 6 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
33		cfzo 13704	. . . . . . 7 class ..^
34	12, 14, 33	co 7420	. . . . . 6 class (0..^(♯‘𝑓))
35	32, 22, 34	wral 3081	. . . . 5 wff ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
36	11, 21, 35	w3a 1103	. . . 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 5175	. . 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 5194	. 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 1570	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 used by:  upwlksfval  48978