Theorem elwwlks2on

Description: A walk of length 2 between two vertices as length 3 string. (Contributed by Alexander van der Vekens, 15-Feb-2018) (Revised by AV, 12-May-2021)

	Ref	Expression
Hypothesis	elwwlks2on.v	|- V = ( Vtx ` G )
Assertion	elwwlks2on	|- ( ( G e. UPGraph /\ A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> E. b e. V ( W = <" A b C "> /\ E. f ( f ( Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elwwlks2on.v	|- V = ( Vtx ` G )
2	1	elwwlks2ons3	|- ( W e. ( A ( 2 WWalksNOn G ) C ) <-> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
3	1	s3wwlks2on	|- ( ( G e. UPGraph /\ A e. V /\ C e. V ) -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> E. f ( f ( Walks ` G ) <" A b C "> /\ ( # ` f ) = 2 ) ) )
4		breq2	|- ( <" A b C "> = W -> ( f ( Walks ` G ) <" A b C "> <-> f ( Walks ` G ) W ) )
5	4	eqcoms	|- ( W = <" A b C "> -> ( f ( Walks ` G ) <" A b C "> <-> f ( Walks ` G ) W ) )
6	5	anbi1d	|- ( W = <" A b C "> -> ( ( f ( Walks ` G ) <" A b C "> /\ ( # ` f ) = 2 ) <-> ( f ( Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
7	6	exbidv	|- ( W = <" A b C "> -> ( E. f ( f ( Walks ` G ) <" A b C "> /\ ( # ` f ) = 2 ) <-> E. f ( f ( Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
8	3 7	sylan9bb	|- ( ( ( G e. UPGraph /\ A e. V /\ C e. V ) /\ W = <" A b C "> ) -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> E. f ( f ( Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
9	8	pm5.32da	|- ( ( G e. UPGraph /\ A e. V /\ C e. V ) -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( W = <" A b C "> /\ E. f ( f ( Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) )
10	9	rexbidv	|- ( ( G e. UPGraph /\ A e. V /\ C e. V ) -> ( E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> E. b e. V ( W = <" A b C "> /\ E. f ( f ( Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) )
11	2 10	syl5bb	|- ( ( G e. UPGraph /\ A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> E. b e. V ( W = <" A b C "> /\ E. f ( f ( Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) )