Metamath Proof Explorer

Theorem cyclprop

Description: The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Assertion cyclprop 
|- ( F ( Cycles ` G ) P -> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )

Proof

Step Hyp Ref Expression
1 iscycl 
 |-  ( F ( Cycles ` G ) P <-> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
2 1 biimpi 
 |-  ( F ( Cycles ` G ) P -> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )