Metamath Proof Explorer


Theorem trlf1

Description: The enumeration F of a trail <. F , P >. is injective. (Contributed by AV, 20-Feb-2021) (Proof shortened by AV, 29-Oct-2021)

Ref Expression
Hypothesis trlf1.i
|- I = ( iEdg ` G )
Assertion trlf1
|- ( F ( Trails ` G ) P -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I )

Proof

Step Hyp Ref Expression
1 trlf1.i
 |-  I = ( iEdg ` G )
2 istrl
 |-  ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) )
3 1 wlkf
 |-  ( F ( Walks ` G ) P -> F e. Word dom I )
4 wrdf
 |-  ( F e. Word dom I -> F : ( 0 ..^ ( # ` F ) ) --> dom I )
5 df-f1
 |-  ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I <-> ( F : ( 0 ..^ ( # ` F ) ) --> dom I /\ Fun `' F ) )
6 5 simplbi2
 |-  ( F : ( 0 ..^ ( # ` F ) ) --> dom I -> ( Fun `' F -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) )
7 3 4 6 3syl
 |-  ( F ( Walks ` G ) P -> ( Fun `' F -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) )
8 7 imp
 |-  ( ( F ( Walks ` G ) P /\ Fun `' F ) -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I )
9 2 8 sylbi
 |-  ( F ( Trails ` G ) P -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I )