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Definition df-pth 23417
Description: Define the set of all Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 23507).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Assertion
Ref Expression
df-pth  |- Paths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
Distinct variable group:    v, e, f, p

Detailed syntax breakdown of Definition df-pth
StepHypRef Expression
1 cpath 23407 . 2  class Paths
2 vv . . 3  setvar  v
3 ve . . 3  setvar  e
4 cvv 2972 . . 3  class  _V
5 vf . . . . . . 7  setvar  f
65cv 1368 . . . . . 6  class  f
7 vp . . . . . . 7  setvar  p
87cv 1368 . . . . . 6  class  p
92cv 1368 . . . . . . 7  class  v
103cv 1368 . . . . . . 7  class  e
11 ctrail 23406 . . . . . . 7  class Trails
129, 10, 11co 6091 . . . . . 6  class  ( v Trails 
e )
136, 8, 12wbr 4292 . . . . 5  wff  f ( v Trails  e ) p
14 c1 9283 . . . . . . . . 9  class  1
15 chash 12103 . . . . . . . . . 10  class  #
166, 15cfv 5418 . . . . . . . . 9  class  ( # `  f )
17 cfzo 11548 . . . . . . . . 9  class ..^
1814, 16, 17co 6091 . . . . . . . 8  class  ( 1..^ ( # `  f
) )
198, 18cres 4842 . . . . . . 7  class  ( p  |`  ( 1..^ ( # `  f ) ) )
2019ccnv 4839 . . . . . 6  class  `' ( p  |`  ( 1..^ ( # `  f
) ) )
2120wfun 5412 . . . . 5  wff  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )
22 cc0 9282 . . . . . . . . 9  class  0
2322, 16cpr 3879 . . . . . . . 8  class  { 0 ,  ( # `  f
) }
248, 23cima 4843 . . . . . . 7  class  ( p
" { 0 ,  ( # `  f
) } )
258, 18cima 4843 . . . . . . 7  class  ( p
" ( 1..^ (
# `  f )
) )
2624, 25cin 3327 . . . . . 6  class  ( ( p " { 0 ,  ( # `  f
) } )  i^i  ( p " (
1..^ ( # `  f
) ) ) )
27 c0 3637 . . . . . 6  class  (/)
2826, 27wceq 1369 . . . . 5  wff  ( ( p " { 0 ,  ( # `  f
) } )  i^i  ( p " (
1..^ ( # `  f
) ) ) )  =  (/)
2913, 21, 28w3a 965 . . . 4  wff  ( f ( v Trails  e ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) )
3029, 5, 7copab 4349 . . 3  class  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) }
312, 3, 4, 4, 30cmpt2 6093 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
321, 31wceq 1369 1  wff Paths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  pths  23465  pthistrl  23471  pthdepisspth  23473
  Copyright terms: Public domain W3C validator