Definition df-upwlks 48050

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 48049	. 2 class UPWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3480	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1539	. . . . . 6 class 𝑓
6	2	cv 1539	. . . . . . . . 9 class 𝑔
7		ciedg 29014	. . . . . . . . 9 class iEdg
8	6, 7	cfv 6561	. . . . . . . 8 class (iEdg‘𝑔)
9	8	cdm 5685	. . . . . . 7 class dom (iEdg‘𝑔)
10	9	cword 14552	. . . . . 6 class Word dom (iEdg‘𝑔)
11	5, 10	wcel 2108	. . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12		cc0 11155	. . . . . . 7 class 0
13		chash 14369	. . . . . . . 8 class ♯
14	5, 13	cfv 6561	. . . . . . 7 class (♯‘𝑓)
15		cfz 13547	. . . . . . 7 class ...
16	12, 14, 15	co 7431	. . . . . 6 class (0...(♯‘𝑓))
17		cvtx 29013	. . . . . . 7 class Vtx
18	6, 17	cfv 6561	. . . . . 6 class (Vtx‘𝑔)
19		vp	. . . . . . 7 setvar 𝑝
20	19	cv 1539	. . . . . 6 class 𝑝
21	16, 18, 20	wf 6557	. . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22		vk	. . . . . . . . . 10 setvar 𝑘
23	22	cv 1539	. . . . . . . . 9 class 𝑘
24	23, 5	cfv 6561	. . . . . . . 8 class (𝑓‘𝑘)
25	24, 8	cfv 6561	. . . . . . 7 class ((iEdg‘𝑔)‘(𝑓‘𝑘))
26	23, 20	cfv 6561	. . . . . . . 8 class (𝑝‘𝑘)
27		c1 11156	. . . . . . . . . 10 class 1
28		caddc 11158	. . . . . . . . . 10 class +
29	23, 27, 28	co 7431	. . . . . . . . 9 class (𝑘 + 1)
30	29, 20	cfv 6561	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
31	26, 30	cpr 4628	. . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
32	25, 31	wceq 1540	. . . . . 6 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
33		cfzo 13694	. . . . . . 7 class ..^
34	12, 14, 33	co 7431	. . . . . 6 class (0..^(♯‘𝑓))
35	32, 22, 34	wral 3061	. . . . 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 5205	. . 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 5225	. 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 1540	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  48051