Definition df-clwwlk 30566

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 30351. 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 30565	. 2 class ClWWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3451	. . 3 class V
4		vw	. . . . . . 7 setvar 𝑤
5	4	cv 1569	. . . . . 6 class 𝑤
6		c0 4279	. . . . . 6 class ∅
7	5, 6	wne 2956	. . . . 5 wff 𝑤 ≠ ∅
8		vi	. . . . . . . . . 10 setvar 𝑖
9	8	cv 1569	. . . . . . . . 9 class 𝑖
10	9, 5	cfv 6537	. . . . . . . 8 class (𝑤‘𝑖)
11		c1 11194	. . . . . . . . . 10 class 1
12		caddc 11196	. . . . . . . . . 10 class +
13	9, 11, 12	co 7418	. . . . . . . . 9 class (𝑖 + 1)
14	13, 5	cfv 6537	. . . . . . . 8 class (𝑤‘(𝑖 + 1))
15	10, 14	cpr 4586	. . . . . . 7 class {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))}
16	2	cv 1569	. . . . . . . 8 class 𝑔
17		cedg 29618	. . . . . . . 8 class Edg
18	16, 17	cfv 6537	. . . . . . 7 class (Edg‘𝑔)
19	15, 18	wcel 2145	. . . . . 6 wff {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
20		cc0 11193	. . . . . . 7 class 0
21		chash 14467	. . . . . . . . 9 class ♯
22	5, 21	cfv 6537	. . . . . . . 8 class (♯‘𝑤)
23		cmin 11534	. . . . . . . 8 class −
24	22, 11, 23	co 7418	. . . . . . 7 class ((♯‘𝑤) − 1)
25		cfzo 13781	. . . . . . 7 class ..^
26	20, 24, 25	co 7418	. . . . . 6 class (0..^((♯‘𝑤) − 1))
27	19, 8, 26	wral 3077	. . . . 5 wff ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
28		clsw 14700	. . . . . . . 8 class lastS
29	5, 28	cfv 6537	. . . . . . 7 class (lastS‘𝑤)
30	20, 5	cfv 6537	. . . . . . 7 class (𝑤‘0)
31	29, 30	cpr 4586	. . . . . 6 class {(lastS‘𝑤), (𝑤‘0)}
32	31, 18	wcel 2145	. . . . 5 wff {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔)
33	7, 27, 32	w3a 1103	. . . 4 wff (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))
34		cvtx 29567	. . . . . 6 class Vtx
35	16, 34	cfv 6537	. . . . 5 class (Vtx‘𝑔)
36	35	cword 14651	. . . 4 class Word (Vtx‘𝑔)
37	33, 4, 36	crab 3413	. . 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 5186	. 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 1570	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:  clwwlk  30567