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Definition df-wwlkn 30326
Description: Define the set of all 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) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlk 23427. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
df-wwlkn  |- WWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
Distinct variable group:    e, n, v, w

Detailed syntax breakdown of Definition df-wwlkn
StepHypRef Expression
1 cwwlkn 30324 . 2  class WWalksN
2 vv . . 3  setvar  v
3 ve . . 3  setvar  e
4 cvv 2984 . . 3  class  _V
5 vn . . . 4  setvar  n
6 cn0 10591 . . . 4  class  NN0
7 vw . . . . . . . 8  setvar  w
87cv 1368 . . . . . . 7  class  w
9 chash 12115 . . . . . . 7  class  #
108, 9cfv 5430 . . . . . 6  class  ( # `  w )
115cv 1368 . . . . . . 7  class  n
12 c1 9295 . . . . . . 7  class  1
13 caddc 9297 . . . . . . 7  class  +
1411, 12, 13co 6103 . . . . . 6  class  ( n  +  1 )
1510, 14wceq 1369 . . . . 5  wff  ( # `  w )  =  ( n  +  1 )
162cv 1368 . . . . . 6  class  v
173cv 1368 . . . . . 6  class  e
18 cwwlk 30323 . . . . . 6  class WWalks
1916, 17, 18co 6103 . . . . 5  class  ( v WWalks 
e )
2015, 7, 19crab 2731 . . . 4  class  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) }
215, 6, 20cmpt 4362 . . 3  class  ( n  e.  NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } )
222, 3, 4, 4, 21cmpt2 6105 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
231, 22wceq 1369 1  wff WWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
Colors of variables: wff setvar class
This definition is referenced by:  wwlkn  30328  wwlknprop  30332
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