Definition df-cycls

Description: Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory) , 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex."

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle." See Definition of Bollobas p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017) (Revised by AV, 31-Jan-2021)

	Ref	Expression
Assertion	df-cycls	|- Cycles = ( g e. _V |-> { <. f , p >. | ( f ( Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ccycls	|- Cycles
1		vg	|- g
2		cvv	|- _V
3		vf	|- f
4		vp	|- p
5	3	cv	|- f
6		cpths	|- Paths
7	1	cv	|- g
8	7 6	cfv	|- ( Paths ` g )
9	4	cv	|- p
10	5 9 8	wbr	|- f ( Paths ` g ) p
11		cc0	|- 0
12	11 9	cfv	|- ( p ` 0 )
13		chash	|- #
14	5 13	cfv	|- ( # ` f )
15	14 9	cfv	|- ( p ` ( # ` f ) )
16	12 15	wceq	|- ( p ` 0 ) = ( p ` ( # ` f ) )
17	10 16	wa	|- ( f ( Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) )
18	17 3 4	copab	|- { <. f , p >. | ( f ( Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }
19	1 2 18	cmpt	|- ( g e. _V |-> { <. f , p >. | ( f ( Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
20	0 19	wceq	|- Cycles = ( g e. _V |-> { <. f , p >. | ( f ( Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )