Definition df-clwwlk 30069

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 in df-clwlks 29856. 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 30068	. 2 class ClWWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3442	. . 3 class V
4		vw	. . . . . . 7 setvar 𝑤
5	4	cv 1541	. . . . . 6 class 𝑤
6		c0 4287	. . . . . 6 class ∅
7	5, 6	wne 2933	. . . . 5 wff 𝑤 ≠ ∅
8		vi	. . . . . . . . . 10 setvar 𝑖
9	8	cv 1541	. . . . . . . . 9 class 𝑖
10	9, 5	cfv 6500	. . . . . . . 8 class (𝑤‘𝑖)
11		c1 11039	. . . . . . . . . 10 class 1
12		caddc 11041	. . . . . . . . . 10 class +
13	9, 11, 12	co 7368	. . . . . . . . 9 class (𝑖 + 1)
14	13, 5	cfv 6500	. . . . . . . 8 class (𝑤‘(𝑖 + 1))
15	10, 14	cpr 4584	. . . . . . 7 class {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))}
16	2	cv 1541	. . . . . . . 8 class 𝑔
17		cedg 29132	. . . . . . . 8 class Edg
18	16, 17	cfv 6500	. . . . . . 7 class (Edg‘𝑔)
19	15, 18	wcel 2114	. . . . . 6 wff {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
20		cc0 11038	. . . . . . 7 class 0
21		chash 14265	. . . . . . . . 9 class ♯
22	5, 21	cfv 6500	. . . . . . . 8 class (♯‘𝑤)
23		cmin 11376	. . . . . . . 8 class −
24	22, 11, 23	co 7368	. . . . . . 7 class ((♯‘𝑤) − 1)
25		cfzo 13582	. . . . . . 7 class ..^
26	20, 24, 25	co 7368	. . . . . 6 class (0..^((♯‘𝑤) − 1))
27	19, 8, 26	wral 3052	. . . . 5 wff ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
28		clsw 14497	. . . . . . . 8 class lastS
29	5, 28	cfv 6500	. . . . . . 7 class (lastS‘𝑤)
30	20, 5	cfv 6500	. . . . . . 7 class (𝑤‘0)
31	29, 30	cpr 4584	. . . . . 6 class {(lastS‘𝑤), (𝑤‘0)}
32	31, 18	wcel 2114	. . . . 5 wff {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔)
33	7, 27, 32	w3a 1087	. . . 4 wff (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))
34		cvtx 29081	. . . . . 6 class Vtx
35	16, 34	cfv 6500	. . . . 5 class (Vtx‘𝑔)
36	35	cword 14448	. . . 4 class Word (Vtx‘𝑔)
37	33, 4, 36	crab 3401	. . 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 5181	. 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 1542	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 referenced by:  clwwlk  30070