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Definition df-clwwlkn 25050
Description: Define the set of all Closed Walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlk 25048. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
Assertion
Ref Expression
df-clwwlkn  |- ClWWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }
) )
Distinct variable group:    e, n, v, w

Detailed syntax breakdown of Definition df-clwwlkn
StepHypRef Expression
1 cclwwlkn 25047 . 2  class ClWWalksN
2 vv . . 3  setvar  v
3 ve . . 3  setvar  e
4 cvv 3058 . . 3  class  _V
5 vn . . . 4  setvar  n
6 cn0 10756 . . . 4  class  NN0
7 vw . . . . . . . 8  setvar  w
87cv 1404 . . . . . . 7  class  w
9 chash 12359 . . . . . . 7  class  #
108, 9cfv 5525 . . . . . 6  class  ( # `  w )
115cv 1404 . . . . . 6  class  n
1210, 11wceq 1405 . . . . 5  wff  ( # `  w )  =  n
132cv 1404 . . . . . 6  class  v
143cv 1404 . . . . . 6  class  e
15 cclwwlk 25046 . . . . . 6  class ClWWalks
1613, 14, 15co 6234 . . . . 5  class  ( v ClWWalks  e )
1712, 7, 16crab 2757 . . . 4  class  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }
185, 6, 17cmpt 4452 . . 3  class  ( n  e.  NN0  |->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }
)
192, 3, 4, 4, 18cmpt2 6236 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }
) )
201, 19wceq 1405 1  wff ClWWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }
) )
Colors of variables: wff setvar class
This definition is referenced by:  clwwlkn  25065  clwwlknprop  25070
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