Definition df-clwwlkn

Description: Define the set of all closed walks of a fixed length n as words over the set of vertices in a graph g . If 0 < n , 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 . For n = 0 , the set is empty, see clwwlkn0 . (Contributed by Alexander van der Vekens, 20-Mar-2018) (Revised by AV, 24-Apr-2021) (Revised by AV, 22-Mar-2022)

	Ref	Expression
Assertion	df-clwwlkn	|- ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. ( ClWWalks ` g ) | ( # ` w ) = n } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cclwwlkn	|- ClWWalksN
1		vn	|- n
2		cn0	|- NN0
3		vg	|- g
4		cvv	|- _V
5		vw	|- w
6		cclwwlk	|- ClWWalks
7	3	cv	|- g
8	7 6	cfv	|- ( ClWWalks ` g )
9		chash	|- #
10	5	cv	|- w
11	10 9	cfv	|- ( # ` w )
12	1	cv	|- n
13	11 12	wceq	|- ( # ` w ) = n
14	13 5 8	crab	|- { w e. ( ClWWalks ` g ) | ( # ` w ) = n }
15	1 3 2 4 14	cmpo	|- ( n e. NN0 , g e. _V |-> { w e. ( ClWWalks ` g ) | ( # ` w ) = n } )
16	0 15	wceq	|- ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. ( ClWWalks ` g ) | ( # ` w ) = n } )