Definition df-wlk 26036

Description: Define the set of all Walks (in an undirected graph).

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). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Assertion

Ref	Expression
df-wlk	⊢ Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})

Distinct variable group:   𝑣,𝑒,𝑘,𝑓,𝑝

Detailed syntax breakdown of Definition df-wlk

Step	Hyp	Ref	Expression
1		cwalk 26026	. 2 class Walks
2		vv	. . 3 setvar 𝑣
3		ve	. . 3 setvar 𝑒
4		cvv 3173	. . 3 class V
5		vf	. . . . . . 7 setvar 𝑓
6	5	cv 1474	. . . . . 6 class 𝑓
7	3	cv 1474	. . . . . . . 8 class 𝑒
8	7	cdm 5038	. . . . . . 7 class dom 𝑒
9	8	cword 13146	. . . . . 6 class Word dom 𝑒
10	6, 9	wcel 1977	. . . . 5 wff 𝑓 ∈ Word dom 𝑒
11		cc0 9815	. . . . . . 7 class 0
12		chash 12979	. . . . . . . 8 class #
13	6, 12	cfv 5804	. . . . . . 7 class (#‘𝑓)
14		cfz 12197	. . . . . . 7 class ...
15	11, 13, 14	co 6549	. . . . . 6 class (0...(#‘𝑓))
16	2	cv 1474	. . . . . 6 class 𝑣
17		vp	. . . . . . 7 setvar 𝑝
18	17	cv 1474	. . . . . 6 class 𝑝
19	15, 16, 18	wf 5800	. . . . 5 wff 𝑝:(0...(#‘𝑓))⟶𝑣
20		vk	. . . . . . . . . 10 setvar 𝑘
21	20	cv 1474	. . . . . . . . 9 class 𝑘
22	21, 6	cfv 5804	. . . . . . . 8 class (𝑓‘𝑘)
23	22, 7	cfv 5804	. . . . . . 7 class (𝑒‘(𝑓‘𝑘))
24	21, 18	cfv 5804	. . . . . . . 8 class (𝑝‘𝑘)
25		c1 9816	. . . . . . . . . 10 class 1
26		caddc 9818	. . . . . . . . . 10 class +
27	21, 25, 26	co 6549	. . . . . . . . 9 class (𝑘 + 1)
28	27, 18	cfv 5804	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
29	24, 28	cpr 4127	. . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
30	23, 29	wceq 1475	. . . . . 6 wff (𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
31		cfzo 12334	. . . . . . 7 class ..^
32	11, 13, 31	co 6549	. . . . . 6 class (0..^(#‘𝑓))
33	30, 20, 32	wral 2896	. . . . 5 wff ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
34	10, 19, 33	w3a 1031	. . . 4 wff (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})
35	34, 5, 17	copab 4642	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}
36	2, 3, 4, 4, 35	cmpt2 6551	. 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
37	1, 36	wceq 1475	1 wff Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})

This definition is referenced by:  relwlk  26046  wlks  26047  2mwlk  26049  wlkbprop  26051  wlkcompim  26054  wlkelwrd  26058  wlkdvspth  26138