Metamath Proof Explorer


Theorem clwlkwlk

Description: Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018) (Revised by AV, 16-Feb-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Assertion clwlkwlk
|- ( W e. ( ClWalks ` G ) -> W e. ( Walks ` G ) )

Proof

Step Hyp Ref Expression
1 elopabran
 |-  ( W e. { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } -> W e. ( Walks ` G ) )
2 clwlks
 |-  ( ClWalks ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }
3 1 2 eleq2s
 |-  ( W e. ( ClWalks ` G ) -> W e. ( Walks ` G ) )