Definition df-clwwlk 30452

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 30237. 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 30451	. 2 class ClWWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3450	. . 3 class V
4		vw	. . . . . . 7 setvar 𝑤
5	4	cv 1569	. . . . . 6 class 𝑤
6		c0 4279	. . . . . 6 class ∅
7	5, 6	wne 2955	. . . . 5 wff 𝑤 ≠ ∅
8		vi	. . . . . . . . . 10 setvar 𝑖
9	8	cv 1569	. . . . . . . . 9 class 𝑖
10	9, 5	cfv 6533	. . . . . . . 8 class (𝑤‘𝑖)
11		c1 11125	. . . . . . . . . 10 class 1
12		caddc 11127	. . . . . . . . . 10 class +
13	9, 11, 12	co 7413	. . . . . . . . 9 class (𝑖 + 1)
14	13, 5	cfv 6533	. . . . . . . 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 29504	. . . . . . . 8 class Edg
18	16, 17	cfv 6533	. . . . . . 7 class (Edg‘𝑔)
19	15, 18	wcel 2145	. . . . . 6 wff {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
20		cc0 11124	. . . . . . 7 class 0
21		chash 14394	. . . . . . . . 9 class ♯
22	5, 21	cfv 6533	. . . . . . . 8 class (♯‘𝑤)
23		cmin 11465	. . . . . . . 8 class −
24	22, 11, 23	co 7413	. . . . . . 7 class ((♯‘𝑤) − 1)
25		cfzo 13709	. . . . . . 7 class ..^
26	20, 24, 25	co 7413	. . . . . 6 class (0..^((♯‘𝑤) − 1))
27	19, 8, 26	wral 3076	. . . . 5 wff ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
28		clsw 14627	. . . . . . . 8 class lastS
29	5, 28	cfv 6533	. . . . . . 7 class (lastS‘𝑤)
30	20, 5	cfv 6533	. . . . . . 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 29453	. . . . . 6 class Vtx
35	16, 34	cfv 6533	. . . . 5 class (Vtx‘𝑔)
36	35	cword 14578	. . . 4 class Word (Vtx‘𝑔)
37	33, 4, 36	crab 3412	. . 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  30453