Metamath Proof Explorer


Theorem crcts

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

Ref Expression
Assertion crcts
|- ( Circuits ` G ) = { <. f , p >. | ( f ( Trails ` 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-crcts
 |-  Circuits = ( g e. _V |-> { <. f , p >. | ( f ( Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
3 1 2 fvmptopab
 |-  ( Circuits ` G ) = { <. f , p >. | ( f ( Trails ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }