Theorem iswwlksn

Description: A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018) (Revised by AV, 8-Apr-2021)

	Ref	Expression
Assertion	iswwlksn	|- ( N e. NN0 -> ( W e. ( N WWalksN G ) <-> ( W e. ( WWalks ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		wwlksn	|- ( N e. NN0 -> ( N WWalksN G ) = { w e. ( WWalks ` G ) | ( # ` w ) = ( N + 1 ) } )
2	1	eleq2d	|- ( N e. NN0 -> ( W e. ( N WWalksN G ) <-> W e. { w e. ( WWalks ` G ) | ( # ` w ) = ( N + 1 ) } ) )
3		fveqeq2	|- ( w = W -> ( ( # ` w ) = ( N + 1 ) <-> ( # ` W ) = ( N + 1 ) ) )
4	3	elrab	|- ( W e. { w e. ( WWalks ` G ) | ( # ` w ) = ( N + 1 ) } <-> ( W e. ( WWalks ` G ) /\ ( # ` W ) = ( N + 1 ) ) )
5	2 4	bitrdi	|- ( N e. NN0 -> ( W e. ( N WWalksN G ) <-> ( W e. ( WWalks ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) )