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 ) ) |
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 ) ) |