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Metamath Proof Explorer

Theorem upgrf1istrl

Description: Properties of a pair of a one-to-one function into the set of indices of edges and a function into the set of vertices to be a trail in a pseudograph. (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 7-Jan-2021) (Revised by AV, 29-Oct-2021)

Ref Expression
Hypotheses upgrtrls.v 
|- V = ( Vtx ` G )
upgrtrls.i 
|- I = ( iEdg ` G )
Assertion upgrf1istrl 
|- ( G e. UPGraph -> ( F ( Trails ` G ) P <-> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )

Proof

Step Hyp Ref Expression
1 upgrtrls.v 
 |-  V = ( Vtx ` G )
2 upgrtrls.i 
 |-  I = ( iEdg ` G )
3 1 2 upgristrl 
 |-  ( G e. UPGraph -> ( F ( Trails ` G ) P <-> ( ( F e. Word dom I /\ Fun `' F ) /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
4 iswrdb 
 |-  ( F e. Word dom I <-> F : ( 0 ..^ ( # ` F ) ) --> dom I )
5 4 a1i 
 |-  ( G e. UPGraph -> ( F e. Word dom I <-> F : ( 0 ..^ ( # ` F ) ) --> dom I ) )
6 5 anbi1d 
 |-  ( G e. UPGraph -> ( ( F e. Word dom I /\ Fun `' F ) <-> ( F : ( 0 ..^ ( # ` F ) ) --> dom I /\ Fun `' F ) ) )
7 df-f1 
 |-  ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I <-> ( F : ( 0 ..^ ( # ` F ) ) --> dom I /\ Fun `' F ) )
8 6 7 bitr4di 
 |-  ( G e. UPGraph -> ( ( F e. Word dom I /\ Fun `' F ) <-> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) )
9 8 3anbi1d 
 |-  ( G e. UPGraph -> ( ( ( F e. Word dom I /\ Fun `' F ) /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) <-> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
10 3 9 bitrd 
 |-  ( G e. UPGraph -> ( F ( Trails ` G ) P <-> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )