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Metamath Proof Explorer

Theorem wlkvtxeledg

Description: Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021) (Proof shortened by AV, 4-Apr-2021)

Ref Expression
Hypothesis wlkvtxeledg.i 
|- I = ( iEdg ` G )
Assertion wlkvtxeledg 
|- ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )

Proof

Step Hyp Ref Expression
1 wlkvtxeledg.i 
 |-  I = ( iEdg ` G )
2 wlkv 
 |-  ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
3 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
4 3 1 iswlk 
 |-  ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( 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 ) ) ) ) ) )
5 ifpsnprss 
 |-  ( if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
6 5 ralimi 
 |-  ( 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 ) ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
7 6 3ad2ant3 
 |-  ( ( 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 ) ) ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
8 4 7 biimtrdi 
 |-  ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
9 2 8 mpcom 
 |-  ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )