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Definition df-wlkon 28324
Description: Define the collection of walks with particular endpoints (in an un- directed graph). This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
Assertion
Ref Expression
df-wlkon  |- WalkOn  =  ( v  e.  _V , 
e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  a  /\  ( p `  ( # `
 f ) )  =  b ) } ) )
Distinct variable groups:    v, e,
f, p    a, b,
e, f, p, v

Detailed syntax breakdown of Definition df-wlkon
StepHypRef Expression
1 cwlkon 28313 . 2  class WalkOn
2 vv . . 3  set  v
3 ve . . 3  set  e
4 cvv 2801 . . 3  class  _V
5 va . . . 4  set  a
6 vb . . . 4  set  b
72cv 1631 . . . 4  class  v
8 vf . . . . . . . 8  set  f
98cv 1631 . . . . . . 7  class  f
10 vp . . . . . . . 8  set  p
1110cv 1631 . . . . . . 7  class  p
123cv 1631 . . . . . . . 8  class  e
13 cwalk 28309 . . . . . . . 8  class Walks
147, 12, 13co 5874 . . . . . . 7  class  ( v Walks 
e )
159, 11, 14wbr 4039 . . . . . 6  wff  f ( v Walks  e ) p
16 cc0 8753 . . . . . . . 8  class  0
1716, 11cfv 5271 . . . . . . 7  class  ( p `
 0 )
185cv 1631 . . . . . . 7  class  a
1917, 18wceq 1632 . . . . . 6  wff  ( p `
 0 )  =  a
20 chash 11353 . . . . . . . . 9  class  #
219, 20cfv 5271 . . . . . . . 8  class  ( # `  f )
2221, 11cfv 5271 . . . . . . 7  class  ( p `
 ( # `  f
) )
236cv 1631 . . . . . . 7  class  b
2422, 23wceq 1632 . . . . . 6  wff  ( p `
 ( # `  f
) )  =  b
2515, 19, 24w3a 934 . . . . 5  wff  ( f ( v Walks  e ) p  /\  ( p `
 0 )  =  a  /\  ( p `
 ( # `  f
) )  =  b )
2625, 8, 10copab 4092 . . . 4  class  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) }
275, 6, 7, 7, 26cmpt2 5876 . . 3  class  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } )
282, 3, 4, 4, 27cmpt2 5876 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } ) )
291, 28wceq 1632 1  wff WalkOn  =  ( v  e.  _V , 
e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  a  /\  ( p `  ( # `
 f ) )  =  b ) } ) )
Colors of variables: wff set class
This definition is referenced by:  iswlkon  28331
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