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Definition df-spth 23418
Description: Define the set of all Simple 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."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple 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, 20-Oct-2017.)

Assertion
Ref Expression
df-spth  |- SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
Distinct variable group:    v, e, f, p

Detailed syntax breakdown of Definition df-spth
StepHypRef Expression
1 cspath 23408 . 2  class SPaths
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
148ccnv 4839 . . . . . 6  class  `' p
1514wfun 5412 . . . . 5  wff  Fun  `' p
1613, 15wa 369 . . . 4  wff  ( f ( v Trails  e ) p  /\  Fun  `' p )
1716, 5, 7copab 4349 . . 3  class  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  Fun  `' p ) }
182, 3, 4, 4, 17cmpt2 6093 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  Fun  `' p ) } )
191, 18wceq 1369 1  wff SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
Colors of variables: wff setvar class
This definition is referenced by:  spths  23466  spthispth  23472
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