Definition df-clwwlksn 26878

Description: Define the set of all closed walks (in an undirected graph) of a fixed length n 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 26667. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

Assertion

Ref	Expression
df-clwwlksn	|- ClWWalksN = ( n e. NN , g e. _V |-> { w e. (ClWWalks ` g ) | ( # ` w ) = n } )

Distinct variable group:   g, n, w

Detailed syntax breakdown of Definition df-clwwlksn

Step	Hyp	Ref	Expression
1		cclwwlksn 26876	. 2 class ClWWalksN
2		vn	. . 3 setvar n
3		vg	. . 3 setvar g
4		cn 11020	. . 3 class NN
5		cvv 3200	. . 3 class _V
6		vw	. . . . . . 7 setvar w
7	6	cv 1482	. . . . . 6 class w
8		chash 13117	. . . . . 6 class #
9	7, 8	cfv 5888	. . . . 5 class ( # ` w )
10	2	cv 1482	. . . . 5 class n
11	9, 10	wceq 1483	. . . 4 wff ( # ` w ) = n
12	3	cv 1482	. . . . 5 class g
13		cclwwlks 26875	. . . . 5 class ClWWalks
14	12, 13	cfv 5888	. . . 4 class (ClWWalks ` g )
15	11, 6, 14	crab 2916	. . 3 class { w e. (ClWWalks ` g ) | ( # ` w ) = n }
16	2, 3, 4, 5, 15	cmpt2 6652	. 2 class ( n e. NN , g e. _V |-> { w e. (ClWWalks ` g ) | ( # ` w ) = n } )
17	1, 16	wceq 1483	1 wff ClWWalksN = ( n e. NN , g e. _V |-> { w e. (ClWWalks ` g ) | ( # ` w ) = n } )

This definition is referenced by:  clwwlksn  26881  clwwlknbp0  26884