Definition df-upwlks 47847

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 47846	. 2 class UPWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3488	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1536	. . . . . 6 class 𝑓
6	2	cv 1536	. . . . . . . . 9 class 𝑔
7		ciedg 29024	. . . . . . . . 9 class iEdg
8	6, 7	cfv 6568	. . . . . . . 8 class (iEdg‘𝑔)
9	8	cdm 5695	. . . . . . 7 class dom (iEdg‘𝑔)
10	9	cword 14556	. . . . . 6 class Word dom (iEdg‘𝑔)
11	5, 10	wcel 2108	. . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12		cc0 11178	. . . . . . 7 class 0
13		chash 14373	. . . . . . . 8 class ♯
14	5, 13	cfv 6568	. . . . . . 7 class (♯‘𝑓)
15		cfz 13561	. . . . . . 7 class ...
16	12, 14, 15	co 7443	. . . . . 6 class (0...(♯‘𝑓))
17		cvtx 29023	. . . . . . 7 class Vtx
18	6, 17	cfv 6568	. . . . . 6 class (Vtx‘𝑔)
19		vp	. . . . . . 7 setvar 𝑝
20	19	cv 1536	. . . . . 6 class 𝑝
21	16, 18, 20	wf 6564	. . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22		vk	. . . . . . . . . 10 setvar 𝑘
23	22	cv 1536	. . . . . . . . 9 class 𝑘
24	23, 5	cfv 6568	. . . . . . . 8 class (𝑓‘𝑘)
25	24, 8	cfv 6568	. . . . . . 7 class ((iEdg‘𝑔)‘(𝑓‘𝑘))
26	23, 20	cfv 6568	. . . . . . . 8 class (𝑝‘𝑘)
27		c1 11179	. . . . . . . . . 10 class 1
28		caddc 11181	. . . . . . . . . 10 class +
29	23, 27, 28	co 7443	. . . . . . . . 9 class (𝑘 + 1)
30	29, 20	cfv 6568	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
31	26, 30	cpr 4650	. . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
32	25, 31	wceq 1537	. . . . . 6 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
33		cfzo 13705	. . . . . . 7 class ..^
34	12, 14, 33	co 7443	. . . . . 6 class (0..^(♯‘𝑓))
35	32, 22, 34	wral 3067	. . . . 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 5228	. . 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 5249	. 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 1537	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  47848