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Metamath Proof Explorer

Theorem clwlkcomp

Description: A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018) (Revised by AV, 17-Feb-2021)

Ref Expression
Hypotheses isclwlke.v 
|- V = ( Vtx ` G )
isclwlke.i 
|- I = ( iEdg ` G )
clwlkcomp.1 
|- F = ( 1st ` W )
clwlkcomp.2 
|- P = ( 2nd ` W )
Assertion clwlkcomp 
|- ( ( G e. X /\ W e. ( S X. T ) ) -> ( W e. ( ClWalks ` G ) <-> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V ) /\ ( 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 isclwlke.v 
 |-  V = ( Vtx ` G )
2 isclwlke.i 
 |-  I = ( iEdg ` G )
3 clwlkcomp.1 
 |-  F = ( 1st ` W )
4 clwlkcomp.2 
 |-  P = ( 2nd ` W )
5 3 eqcomi 
 |-  ( 1st ` W ) = F
6 4 eqcomi 
 |-  ( 2nd ` W ) = P
7 5 6 pm3.2i 
 |-  ( ( 1st ` W ) = F /\ ( 2nd ` W ) = P )
8 eqop 
 |-  ( W e. ( S X. T ) -> ( W = <. F , P >. <-> ( ( 1st ` W ) = F /\ ( 2nd ` W ) = P ) ) )
9 7 8 mpbiri 
 |-  ( W e. ( S X. T ) -> W = <. F , P >. )
10 9 eleq1d 
 |-  ( W e. ( S X. T ) -> ( W e. ( ClWalks ` G ) <-> <. F , P >. e. ( ClWalks ` G ) ) )
11 df-br 
 |-  ( F ( ClWalks ` G ) P <-> <. F , P >. e. ( ClWalks ` G ) )
12 10 11 bitr4di 
 |-  ( W e. ( S X. T ) -> ( W e. ( ClWalks ` G ) <-> F ( ClWalks ` G ) P ) )
13 1 2 isclwlke 
 |-  ( G e. X -> ( F ( ClWalks ` G ) P <-> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V ) /\ ( 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) )
14 12 13 sylan9bbr 
 |-  ( ( G e. X /\ W e. ( S X. T ) ) -> ( W e. ( ClWalks ` G ) <-> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V ) /\ ( 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) )