Metamath Proof Explorer


Theorem wlkf

Description: The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021)

Ref Expression
Hypothesis wlkf.i
|- I = ( iEdg ` G )
Assertion wlkf
|- ( F ( Walks ` G ) P -> F e. Word dom I )

Proof

Step Hyp Ref Expression
1 wlkf.i
 |-  I = ( iEdg ` G )
2 eqid
 |-  ( Vtx ` G ) = ( Vtx ` G )
3 2 1 wlkprop
 |-  ( F ( Walks ` G ) P -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
4 3 simp1d
 |-  ( F ( Walks ` G ) P -> F e. Word dom I )