Metamath Proof Explorer


Theorem trlsfval

Description: The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 28-Dec-2020) (Revised by AV, 29-Oct-2021)

Ref Expression
Assertion trlsfval
|- ( Trails ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ Fun `' f ) }

Proof

Step Hyp Ref Expression
1 biidd
 |-  ( g = G -> ( Fun `' f <-> Fun `' f ) )
2 df-trls
 |-  Trails = ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ Fun `' f ) } )
3 1 2 fvmptopab
 |-  ( Trails ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ Fun `' f ) }