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 ) )

Step

Hyp

Ref

Expression

elopabran

 |-  ( W e. { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } -> W e. ( Walks ` G ) )

clwlks

 |-  ( ClWalks ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }

1 2

eleq2s

 |-  ( W e. ( ClWalks ` G ) -> W e. ( Walks ` G ) )