Theorem cycls

Description: The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 31-Jan-2021)

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

Proof

Step	Hyp	Ref	Expression
1		biidd	|- ( ( T. /\ g = G ) -> ( ( p ` 0 ) = ( p ` ( # ` f ) ) <-> ( p ` 0 ) = ( p ` ( # ` f ) ) ) )
2		wksv	|- { <. f , p >. | f ( Walks ` G ) p } e. _V
3		pthiswlk	|- ( f ( Paths ` G ) p -> f ( Walks ` G ) p )
4	3	ssopab2i	|- { <. f , p >. | f ( Paths ` G ) p } C_ { <. f , p >. | f ( Walks ` G ) p }
5	2 4	ssexi	|- { <. f , p >. | f ( Paths ` G ) p } e. _V
6	5	a1i	|- ( T. -> { <. f , p >. | f ( Paths ` G ) p } e. _V )
7		df-cycls	|- Cycles = ( g e. _V |-> { <. f , p >. | ( f ( Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
8	1 6 7	fvmptopab	|- ( T. -> ( Cycles ` G ) = { <. f , p >. | ( f ( Paths ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
9	8	mptru	|- ( Cycles ` G ) = { <. f , p >. | ( f ( Paths ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }