Definition df-clwwlk 29966

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 29753. 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 29965	. 2 class ClWWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3437	. . 3 class V
4		vw	. . . . . . 7 setvar 𝑤
5	4	cv 1540	. . . . . 6 class 𝑤
6		c0 4282	. . . . . 6 class ∅
7	5, 6	wne 2929	. . . . 5 wff 𝑤 ≠ ∅
8		vi	. . . . . . . . . 10 setvar 𝑖
9	8	cv 1540	. . . . . . . . 9 class 𝑖
10	9, 5	cfv 6488	. . . . . . . 8 class (𝑤‘𝑖)
11		c1 11016	. . . . . . . . . 10 class 1
12		caddc 11018	. . . . . . . . . 10 class +
13	9, 11, 12	co 7354	. . . . . . . . 9 class (𝑖 + 1)
14	13, 5	cfv 6488	. . . . . . . 8 class (𝑤‘(𝑖 + 1))
15	10, 14	cpr 4579	. . . . . . 7 class {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))}
16	2	cv 1540	. . . . . . . 8 class 𝑔
17		cedg 29029	. . . . . . . 8 class Edg
18	16, 17	cfv 6488	. . . . . . 7 class (Edg‘𝑔)
19	15, 18	wcel 2113	. . . . . 6 wff {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
20		cc0 11015	. . . . . . 7 class 0
21		chash 14241	. . . . . . . . 9 class ♯
22	5, 21	cfv 6488	. . . . . . . 8 class (♯‘𝑤)
23		cmin 11353	. . . . . . . 8 class −
24	22, 11, 23	co 7354	. . . . . . 7 class ((♯‘𝑤) − 1)
25		cfzo 13558	. . . . . . 7 class ..^
26	20, 24, 25	co 7354	. . . . . 6 class (0..^((♯‘𝑤) − 1))
27	19, 8, 26	wral 3048	. . . . 5 wff ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
28		clsw 14473	. . . . . . . 8 class lastS
29	5, 28	cfv 6488	. . . . . . 7 class (lastS‘𝑤)
30	20, 5	cfv 6488	. . . . . . 7 class (𝑤‘0)
31	29, 30	cpr 4579	. . . . . 6 class {(lastS‘𝑤), (𝑤‘0)}
32	31, 18	wcel 2113	. . . . 5 wff {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔)
33	7, 27, 32	w3a 1086	. . . 4 wff (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))
34		cvtx 28978	. . . . . 6 class Vtx
35	16, 34	cfv 6488	. . . . 5 class (Vtx‘𝑔)
36	35	cword 14424	. . . 4 class Word (Vtx‘𝑔)
37	33, 4, 36	crab 3396	. . 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 5176	. 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 1541	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  29967