Theorem clwlks

Description: The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018) (Revised by AV, 16-Feb-2021) (Revised by AV, 29-Oct-2021)

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

Proof

Step	Hyp	Ref	Expression
1		biidd	|- ( ( T. /\ g = G ) -> ( ( p ` 0 ) = ( p ` ( # ` f ) ) <-> ( p ` 0 ) = ( p ` ( # ` f ) ) ) )
2		wksv	|- { <. f , p >. | f ( Walks ` G ) p } e. _V
3	2	a1i	|- ( T. -> { <. f , p >. | f ( Walks ` G ) p } e. _V )
4		df-clwlks	|- ClWalks = ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
5	1 3 4	fvmptopab	|- ( T. -> ( ClWalks ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
6	5	mptru	|- ( ClWalks ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }