Definition df-clwwlk 16799

Description: Define the set of all closed walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined elsewhere. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

Assertion

Ref	Expression
df-clwwlk	⊢ ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))})

Distinct variable group:   𝑔,𝑖,𝑤

Detailed syntax breakdown of Definition df-clwwlk

Step	Hyp	Ref	Expression
1		cclwwlk 16798	. 2 class ClWWalks
2		vg	. . 3 setvar 𝑔
3		cvv 2821	. . 3 class V
4		vw	. . . . . . 7 setvar 𝑤
5	4	cv 1401	. . . . . 6 class 𝑤
6		c0 3520	. . . . . 6 class ∅
7	5, 6	wne 2420	. . . . 5 wff 𝑤 ≠ ∅
8		vi	. . . . . . . . . 10 setvar 𝑖
9	8	cv 1401	. . . . . . . . 9 class 𝑖
10	9, 5	cfv 5377	. . . . . . . 8 class (𝑤‘𝑖)
11		c1 8181	. . . . . . . . . 10 class 1
12		caddc 8183	. . . . . . . . . 10 class +
13	9, 11, 12	co 6085	. . . . . . . . 9 class (𝑖 + 1)
14	13, 5	cfv 5377	. . . . . . . 8 class (𝑤‘(𝑖 + 1))
15	10, 14	cpr 3710	. . . . . . 7 class {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))}
16	2	cv 1401	. . . . . . . 8 class 𝑔
17		cedg 16464	. . . . . . . 8 class Edg
18	16, 17	cfv 5377	. . . . . . 7 class (Edg‘𝑔)
19	15, 18	wcel 2209	. . . . . 6 wff {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
20		cc0 8180	. . . . . . 7 class 0
21		chash 11230	. . . . . . . . 9 class ♯
22	5, 21	cfv 5377	. . . . . . . 8 class (♯‘𝑤)
23		cmin 8499	. . . . . . . 8 class −
24	22, 11, 23	co 6085	. . . . . . 7 class ((♯‘𝑤) − 1)
25		cfzo 10560	. . . . . . 7 class ..^
26	20, 24, 25	co 6085	. . . . . 6 class (0..^((♯‘𝑤) − 1))
27	19, 8, 26	wral 2528	. . . . 5 wff ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
28		clsw 11365	. . . . . . . 8 class lastS
29	5, 28	cfv 5377	. . . . . . 7 class (lastS‘𝑤)
30	20, 5	cfv 5377	. . . . . . 7 class (𝑤‘0)
31	29, 30	cpr 3710	. . . . . 6 class {(lastS‘𝑤), (𝑤‘0)}
32	31, 18	wcel 2209	. . . . 5 wff {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔)
33	7, 27, 32	w3a 1009	. . . 4 wff (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))
34		cvtx 16419	. . . . . 6 class Vtx
35	16, 34	cfv 5377	. . . . 5 class (Vtx‘𝑔)
36	35	cword 11320	. . . 4 class Word (Vtx‘𝑔)
37	33, 4, 36	crab 2532	. . 3 class {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))}
38	2, 3, 37	cmpt 4192	. 2 class (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))})
39	1, 38	wceq 1402	1 wff ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))})

This definition is used by:  clwwlkg  16800  isclwwlk  16801  clwwlkbp  16802