Definition df-clwwlk 16547

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 elsewhere. 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 e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } )

Distinct variable group:   g, i, w

Detailed syntax breakdown of Definition df-clwwlk

Step	Hyp	Ref	Expression
1		cclwwlk 16546	. 2 class ClWWalks
2		vg	. . 3 setvar g
3		cvv 2821	. . 3 class _V
4		vw	. . . . . . 7 setvar w
5	4	cv 1401	. . . . . 6 class w
6		c0 3520	. . . . . 6 class (/)
7	5, 6	wne 2420	. . . . 5 wff w =/= (/)
8		vi	. . . . . . . . . 10 setvar i
9	8	cv 1401	. . . . . . . . 9 class i
10	9, 5	cfv 5372	. . . . . . . 8 class ( w ` i )
11		c1 8170	. . . . . . . . . 10 class 1
12		caddc 8172	. . . . . . . . . 10 class +
13	9, 11, 12	co 6075	. . . . . . . . 9 class ( i + 1 )
14	13, 5	cfv 5372	. . . . . . . 8 class ( w ` ( i + 1 ) )
15	10, 14	cpr 3706	. . . . . . 7 class { ( w ` i ) , ( w ` ( i + 1 ) ) }
16	2	cv 1401	. . . . . . . 8 class g
17		cedg 16212	. . . . . . . 8 class Edg
18	16, 17	cfv 5372	. . . . . . 7 class (Edg ` g )
19	15, 18	wcel 2209	. . . . . 6 wff { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g )
20		cc0 8169	. . . . . . 7 class 0
21		chash 11192	. . . . . . . . 9 class ♯
22	5, 21	cfv 5372	. . . . . . . 8 class (♯ ` w )
23		cmin 8487	. . . . . . . 8 class -
24	22, 11, 23	co 6075	. . . . . . 7 class ( (♯ ` w ) - 1 )
25		cfzo 10527	. . . . . . 7 class ..^
26	20, 24, 25	co 6075	. . . . . 6 class ( 0..^ ( (♯ ` w ) - 1 ) )
27	19, 8, 26	wral 2528	. . . . 5 wff A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g )
28		clsw 11327	. . . . . . . 8 class lastS
29	5, 28	cfv 5372	. . . . . . 7 class (lastS ` w )
30	20, 5	cfv 5372	. . . . . . 7 class ( w ` 0 )
31	29, 30	cpr 3706	. . . . . 6 class { (lastS ` w ) , ( w ` 0 ) }
32	31, 18	wcel 2209	. . . . 5 wff { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g )
33	7, 27, 32	w3a 1009	. . . 4 wff ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) )
34		cvtx 16167	. . . . . 6 class Vtx
35	16, 34	cfv 5372	. . . . 5 class (Vtx ` g )
36	35	cword 11282	. . . 4 class Word (Vtx ` g )
37	33, 4, 36	crab 2532	. . 3 class { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) }
38	2, 3, 37	cmpt 4187	. 2 class ( g e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } )
39	1, 38	wceq 1402	1 wff ClWWalks = ( g e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } )

This definition is referenced by:  clwwlkg  16548  isclwwlk  16549  clwwlkbp  16550