Definition df-upwlks 48633

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 48632	. 2 class UPWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3431	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1546	. . . . . 6 class 𝑓
6	2	cv 1546	. . . . . . . . 9 class 𝑔
7		ciedg 29085	. . . . . . . . 9 class iEdg
8	6, 7	cfv 6486	. . . . . . . 8 class (iEdg‘𝑔)
9	8	cdm 5619	. . . . . . 7 class dom (iEdg‘𝑔)
10	9	cword 14467	. . . . . 6 class Word dom (iEdg‘𝑔)
11	5, 10	wcel 2119	. . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12		cc0 11030	. . . . . . 7 class 0
13		chash 14284	. . . . . . . 8 class ♯
14	5, 13	cfv 6486	. . . . . . 7 class (♯‘𝑓)
15		cfz 13453	. . . . . . 7 class ...
16	12, 14, 15	co 7357	. . . . . 6 class (0...(♯‘𝑓))
17		cvtx 29084	. . . . . . 7 class Vtx
18	6, 17	cfv 6486	. . . . . 6 class (Vtx‘𝑔)
19		vp	. . . . . . 7 setvar 𝑝
20	19	cv 1546	. . . . . 6 class 𝑝
21	16, 18, 20	wf 6482	. . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22		vk	. . . . . . . . . 10 setvar 𝑘
23	22	cv 1546	. . . . . . . . 9 class 𝑘
24	23, 5	cfv 6486	. . . . . . . 8 class (𝑓‘𝑘)
25	24, 8	cfv 6486	. . . . . . 7 class ((iEdg‘𝑔)‘(𝑓‘𝑘))
26	23, 20	cfv 6486	. . . . . . . 8 class (𝑝‘𝑘)
27		c1 11031	. . . . . . . . . 10 class 1
28		caddc 11033	. . . . . . . . . 10 class +
29	23, 27, 28	co 7357	. . . . . . . . 9 class (𝑘 + 1)
30	29, 20	cfv 6486	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
31	26, 30	cpr 4558	. . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
32	25, 31	wceq 1547	. . . . . 6 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
33		cfzo 13600	. . . . . . 7 class ..^
34	12, 14, 33	co 7357	. . . . . 6 class (0..^(♯‘𝑓))
35	32, 22, 34	wral 3053	. . . . 5 wff ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
36	11, 21, 35	w3a 1092	. . . 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 5135	. . 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 5154	. 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 1547	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  48634