Theorem clwlkcompim

Description: Implications for the properties of the components of a closed walk. (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	clwlkcompim	|- ( 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		elfvex	|- ( W e. ( ClWalks ` G ) -> G e. _V )
6		clwlks	|- ( ClWalks ` G ) = { <. f , g >. | ( f ( Walks ` G ) g /\ ( g ` 0 ) = ( g ` ( # ` f ) ) ) }
7	6	a1i	|- ( G e. _V -> ( ClWalks ` G ) = { <. f , g >. | ( f ( Walks ` G ) g /\ ( g ` 0 ) = ( g ` ( # ` f ) ) ) } )
8	7	eleq2d	|- ( G e. _V -> ( W e. ( ClWalks ` G ) <-> W e. { <. f , g >. | ( f ( Walks ` G ) g /\ ( g ` 0 ) = ( g ` ( # ` f ) ) ) } ) )
9		elopaelxp	|- ( W e. { <. f , g >. | ( f ( Walks ` G ) g /\ ( g ` 0 ) = ( g ` ( # ` f ) ) ) } -> W e. ( _V X. _V ) )
10	9	anim2i	|- ( ( G e. _V /\ W e. { <. f , g >. | ( f ( Walks ` G ) g /\ ( g ` 0 ) = ( g ` ( # ` f ) ) ) } ) -> ( G e. _V /\ W e. ( _V X. _V ) ) )
11	10	ex	|- ( G e. _V -> ( W e. { <. f , g >. | ( f ( Walks ` G ) g /\ ( g ` 0 ) = ( g ` ( # ` f ) ) ) } -> ( G e. _V /\ W e. ( _V X. _V ) ) ) )
12	8 11	sylbid	|- ( G e. _V -> ( W e. ( ClWalks ` G ) -> ( G e. _V /\ W e. ( _V X. _V ) ) ) )
13	5 12	mpcom	|- ( W e. ( ClWalks ` G ) -> ( G e. _V /\ W e. ( _V X. _V ) ) )
14	1 2 3 4	clwlkcomp	|- ( ( G e. _V /\ W e. ( _V X. _V ) ) -> ( 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 ) ) ) ) ) )
15	13 14	syl	|- ( W e. ( ClWalks ` G ) -> ( 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 ) ) ) ) ) )
16	15	ibi	|- ( 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 ) ) ) ) )