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 } )

Metamath Proof Explorer

Definition df-clwwlkn

Detailed syntax breakdown