Metamath Proof Explorer


Theorem pthistrl

Description: A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017) (Revised by AV, 9-Jan-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Assertion pthistrl
|- ( F ( Paths ` G ) P -> F ( Trails ` G ) P )

Proof

Step Hyp Ref Expression
1 ispth
 |-  ( F ( Paths ` G ) P <-> ( F ( Trails ` G ) P /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) )
2 1 simp1bi
 |-  ( F ( Paths ` G ) P -> F ( Trails ` G ) P )