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	|- ( g = G -> ( ( p ` 0 ) = ( p ` ( # ` f ) ) <-> ( p ` 0 ) = ( p ` ( # ` f ) ) ) )
2		df-cycls	|- Cycles = ( g e. _V |-> { <. f , p >. | ( f ( Paths ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
3	1 2	fvmptopab	|- ( Cycles ` G ) = { <. f , p >. | ( f ( Paths ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }