Definition df-clwlk 26278

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

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed 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)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 26293! (Contributed by Alexander van der Vekens, 12-Mar-2018.)

Assertion

Ref	Expression
df-clwlk	⊢ ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-clwlk

Step	Hyp	Ref	Expression
1		cclwlk 26275	. 2 class ClWalks
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		vp	. . . . . . 7 setvar 𝑝
8	7	cv 1474	. . . . . 6 class 𝑝
9	2	cv 1474	. . . . . . 7 class 𝑣
10	3	cv 1474	. . . . . . 7 class 𝑒
11		cwalk 26026	. . . . . . 7 class Walks
12	9, 10, 11	co 6549	. . . . . 6 class (𝑣 Walks 𝑒)
13	6, 8, 12	wbr 4583	. . . . 5 wff 𝑓(𝑣 Walks 𝑒)𝑝
14		cc0 9815	. . . . . . 7 class 0
15	14, 8	cfv 5804	. . . . . 6 class (𝑝‘0)
16		chash 12979	. . . . . . . 8 class #
17	6, 16	cfv 5804	. . . . . . 7 class (#‘𝑓)
18	17, 8	cfv 5804	. . . . . 6 class (𝑝‘(#‘𝑓))
19	15, 18	wceq 1475	. . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
20	13, 19	wa 383	. . . 4 wff (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
21	20, 5, 7	copab 4642	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
22	2, 3, 4, 4, 21	cmpt2 6551	. 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
23	1, 22	wceq 1475	1 wff ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

This definition is referenced by:  clwlk  26281  isclwlkg  26283  clwlkiswlk  26285  clwlkswlks  26286  clwlkcompim  26292