Definition df-clwwlk 29202

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 28995. 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 29201	. 2 class ClWWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3475	. . 3 class V
4		vw	. . . . . . 7 setvar 𝑤
5	4	cv 1541	. . . . . 6 class 𝑤
6		c0 4320	. . . . . 6 class ∅
7	5, 6	wne 2941	. . . . 5 wff 𝑤 ≠ ∅
8		vi	. . . . . . . . . 10 setvar 𝑖
9	8	cv 1541	. . . . . . . . 9 class 𝑖
10	9, 5	cfv 6535	. . . . . . . 8 class (𝑤‘𝑖)
11		c1 11098	. . . . . . . . . 10 class 1
12		caddc 11100	. . . . . . . . . 10 class +
13	9, 11, 12	co 7396	. . . . . . . . 9 class (𝑖 + 1)
14	13, 5	cfv 6535	. . . . . . . 8 class (𝑤‘(𝑖 + 1))
15	10, 14	cpr 4626	. . . . . . 7 class {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))}
16	2	cv 1541	. . . . . . . 8 class 𝑔
17		cedg 28274	. . . . . . . 8 class Edg
18	16, 17	cfv 6535	. . . . . . 7 class (Edg‘𝑔)
19	15, 18	wcel 2107	. . . . . 6 wff {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
20		cc0 11097	. . . . . . 7 class 0
21		chash 14277	. . . . . . . . 9 class ♯
22	5, 21	cfv 6535	. . . . . . . 8 class (♯‘𝑤)
23		cmin 11431	. . . . . . . 8 class −
24	22, 11, 23	co 7396	. . . . . . 7 class ((♯‘𝑤) − 1)
25		cfzo 13614	. . . . . . 7 class ..^
26	20, 24, 25	co 7396	. . . . . 6 class (0..^((♯‘𝑤) − 1))
27	19, 8, 26	wral 3062	. . . . 5 wff ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)
28		clsw 14499	. . . . . . . 8 class lastS
29	5, 28	cfv 6535	. . . . . . 7 class (lastS‘𝑤)
30	20, 5	cfv 6535	. . . . . . 7 class (𝑤‘0)
31	29, 30	cpr 4626	. . . . . 6 class {(lastS‘𝑤), (𝑤‘0)}
32	31, 18	wcel 2107	. . . . 5 wff {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔)
33	7, 27, 32	w3a 1088	. . . 4 wff (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))
34		cvtx 28223	. . . . . 6 class Vtx
35	16, 34	cfv 6535	. . . . 5 class (Vtx‘𝑔)
36	35	cword 14451	. . . 4 class Word (Vtx‘𝑔)
37	33, 4, 36	crab 3433	. . 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 5227	. 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  29203