Definition df-clwwlk 16633

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 16632	. 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 8180	. . . . . . . . . 10 class 1
12		caddc 8182	. . . . . . . . . 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 16298	. . . . . . . 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 8179	. . . . . . 7 class 0
21		chash 11214	. . . . . . . . 9 class ♯
22	5, 21	cfv 5377	. . . . . . . 8 class (♯‘𝑤)
23		cmin 8497	. . . . . . . 8 class −
24	22, 11, 23	co 6085	. . . . . . 7 class ((♯‘𝑤) − 1)
25		cfzo 10549	. . . . . . 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 11349	. . . . . . . 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 16253	. . . . . 6 class Vtx
35	16, 34	cfv 5377	. . . . 5 class (Vtx‘𝑔)
36	35	cword 11304	. . . 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  16634  isclwwlk  16635  clwwlkbp  16636