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Theorem List for Metamath Proof Explorer - 27701-27800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremistrl2 27701* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Trails  E ) P 
 <->  ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P : ( 0
 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theoremtrliswlk 27702 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
 
Theoremtrlon 27703* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( A ( V TrailOn  E ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Trails  E ) p ) } )
 
Theoremistrlon 27704 Properties of a pair of functions to be a trail between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V TrailOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Trails  E ) P ) ) )
 
Theoremtrlonprop 27705 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  ( F ( A ( V TrailOn  E ) B ) P  ->  ( (
 ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V ) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Trails  E ) P ) ) )
 
Theoremtrlonistrl 27706 A trail between to vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
 |-  ( F ( A ( V TrailOn  E ) B ) P  ->  F ( V Trails  E ) P )
 
Theoremtrlonwlkon 27707 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
 |-  ( F ( A ( V TrailOn  E ) B ) P  ->  F ( A ( V WalkOn  E ) B ) P )
 
Theorem0wlk 27708 A pair of an empty set (of edges) and a second set (of vertices) is a walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0trl 27709 A pair of an empty set (of edges) and a second set (of vertices) is a trail if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0wlkon 27710 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  ->  ( ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N )  ->  (/) ( N ( V WalkOn  E ) N ) P ) )
 
Theorem0trlon 27711 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  ->  ( ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N )  ->  (/) ( N ( V TrailOn  E ) N ) P ) )
 
Theoremwlkntrllem1 27712 Lemma 1 for wlkntrl 27717: F is a word over  {
0 }, the domain of E. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  F  e. Word  dom  E
 
Theoremwlkntrllem2 27713 Lemma 2 for wlkntrl 27717: The cardinality of F is 2. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  ( # `  F )  =  2
 
Theoremwlkntrllem3 27714 Lemma 3 for wlkntrl 27717: P is a function on  (
0 ... 2 ) into  { x ,  y }. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  P : ( 0 ... ( # `  F ) ) --> V
 
Theoremwlkntrllem4 27715* Lemma 4 for wlkntrl 27717: The values of E after F are edges between two vertices enumerated by P. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 
Theoremwlkntrllem5 27716* Lemma 5 for wlkntrl 27717: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  -. 
 Fun  `' F
 
Theoremwlkntrl 27717* A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one edge to the other is a walk, but not a trail. Notice that  <. V ,  E >. is a simple graph (without loops) only if  x  =/=  y. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  ( F ( V Walks  E ) P  /\  -.  F ( V Trails  E ) P )
 
Theoremusgrnloop 27718* In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1
 ) ) )
 
Theorempths 27719* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Paths  E )  =  { <. f ,  p >.  |  (
 f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
 ) ) )  /\  ( ( p " { 0 ,  ( # `
  f ) }
 )  i^i  ( p " ( 1..^ ( # `  f ) ) ) )  =  (/) ) }
 )
 
Theoremspths 27720* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V SPaths  E )  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' p ) } )
 
Theoremispth 27721 Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Paths  E ) P  <->  ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) )  /\  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/) ) ) )
 
Theoremisspth 27722 Properties of a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V SPaths  E ) P 
 <->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
 
Theorem0pth 27723 A pair of an empty set (of edges) and a second set (of vertices) is a path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Paths  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0spth 27724 A pair of an empty set (of edges) and a second set (of vertices) is a simple path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V SPaths  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorempthistrl 27725 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
 
Theoremspthispth 27726 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )
 
Theorempthdepisspth 27727 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  (
 ( F ( V Paths  E ) P  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  F ( V SPaths  E ) P )
 
Theorempthon 27728* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( A ( V PathOn  E ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Paths  E ) p ) } )
 
Theoremispthon 27729 Properties of a pair of functions to be a path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V PathOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Paths  E ) P ) ) )
 
Theorempthonprop 27730 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
 |-  ( F ( A ( V PathOn  E ) B ) P  ->  ( (
 ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V ) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Paths  E ) P ) ) )
 
Theorempthonispth 27731 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
 |-  ( F ( A ( V PathOn  E ) B ) P  ->  F ( V Paths  E ) P )
 
Theorem0pthon 27732 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  ->  ( ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N )  ->  (/) ( N ( V PathOn  E ) N ) P ) )
 
Theorem0pthon1 27733 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  ->  (/) ( N ( V PathOn  E ) N ) { <. 0 ,  N >. } )
 
Theorem0pthonv 27734* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( N  e.  V  ->  E. f E. p  f ( N ( V PathOn  E ) N ) p ) )
 
Theoremconstr1trl 27735 Construction of a trail from one given edge in a graph. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )  /\  ( E `  i
 )  =  { A ,  B } )  ->  F ( V Trails  E ) P )
 
Theorem1pthonlem1 27736 Lemma 1 for 1pthon 27738. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. }   =>    |-  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) )
 
Theorem1pthonlem2 27737 Lemma 2 for 1pthon 27738. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. }   =>    |-  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/)
 
Theorem1pthon 27738 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )  /\  ( E `  i )  =  { A ,  B }
 )  ->  { <. 0 ,  i >. }  ( A ( V PathOn  E ) B ) { <. 0 ,  A >. ,  <. 1 ,  B >. } )
 
Theorem1pthoncl 27739 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )  /\  ( I  e. 
 _V  /\  ( E `  I )  =  { A ,  B }
 ) )  ->  { <. 0 ,  I >. }  ( A ( V PathOn  E ) B ) { <. 0 ,  A >. ,  <. 1 ,  B >. } )
 
Theorem1pthon2v 27740* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )  /\  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B } )  ->  E. f E. p  f ( A ( V PathOn  E ) B ) p )
 
Theorem2trllem1 27741 Lemma 1 for constr2trl 27745. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( # `  F )  =  2
 
Theorem2trllem2 27742 Lemma 2 for constr2trl 27745. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( 0..^ ( # `  F ) )  =  {
 0 ,  1 }
 
Theorem2trllem3 27743 Lemma 3 for constr2trl 27745. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( 1..^ ( # `  F ) )  =  {
 1 }
 
Theorem2trllem4 27744 Lemma 4 for constr2trl 27745. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 ->  P  Fn  { 0 ,  1 ,  2 } )
 
Theoremconstr2trl 27745 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C ) 
 ->  ( ( i  =/=  j  /\  ( E `
  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
 )  ->  F ( V Trails  E ) P ) )
 
Theorem2pthonlem1 27746 Lemma 1 for 2pthon 27749. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 ->  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) ) )
 
Theorem2pthonlem2 27747 Lemma 2 for 2pthon 27749. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  B  =/=  C ) )  ->  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/) )
 
Theoremconstr2pth 27748 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. ,  <. 1 ,  j >. }   &    |-  P  =  { <. 0 ,  A >. , 
 <. 1 ,  B >. , 
 <. 2 ,  C >. }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 ->  ( ( i  =/=  j  /\  ( E `
  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
 )  ->  F ( V Paths  E ) P ) )
 
Theorem2pthon 27749 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( (
 i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `
  j )  =  { B ,  C } )  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )
 )
 
Theorem2pthoncl 27750 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  (
 ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 /\  ( I  e. 
 _V  /\  J  e.  _V )  /\  ( I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
 ) )  ->  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )
 
Theorem2pthon3v 27751* For a vertex adjacent to two other vertices there is a path of length 2 between these other vertices. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  (
 ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 /\  ( ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
 )  /\  ( E `  ( `' E `  { A ,  B }
 ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )  ->  E. f E. p ( f ( A ( V PathOn  E ) C ) p  /\  ( # `  f )  =  2 ) )
 
Theoremredwlklem 27752 Lemma for redwlk 27753. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  (
 ( F ( V Walks  E ) P  /\  1  <_  ( # `  F ) )  ->  ( # `  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) )  =  ( ( # `  F )  -  1 ) )
 
Theoremredwlk 27753 A walk ending at the last but one vertex of the walk is a walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  (
 ( F ( V Walks  E ) P  /\  1  <_  ( # `  F ) )  ->  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) ( V Walks  E ) ( P  |`  ( 0..^ ( # `  F ) ) ) )
 
Theoremwlkdvspthlem 27754* Lemma for wlkdvspth 27755. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
 |-  (
 ( F  e. Word  dom  E 
 /\  P : ( 0 ... ( # `  F ) ) -1-1-> V  /\  A. k  e.  (
 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 )  ->  Fun  `' F )
 
Theoremwlkdvspth 27755 A walk consisting of different vertices is a simple path. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
 |-  (
 ( F ( V Walks  E ) P  /\  Fun  `' P )  ->  F ( V SPaths  E ) P )
 
Theoremcrcts 27756* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Circuits  E )  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  ( p `  0 )  =  ( p `  ( # `  f ) ) ) } )
 
Theoremcycls 27757* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Cycles  E )  =  { <. f ,  p >.  |  ( f ( V Paths  E ) p 
 /\  ( p `  0 )  =  ( p `  ( # `  f
 ) ) ) }
 )
 
Theoremiscrct 27758 Properties of a pair of functions to be a circuit (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Circuits  E ) P 
 <->  ( F ( V Trails  E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
Theoremiscycl 27759 Properties of a pair of functions to be a cycle (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Cycles  E ) P 
 <->  ( F ( V Paths  E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
Theorem0crct 27760 A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Circuits  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0cycl 27761 A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Cycles  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theoremcrctistrl 27762 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F ( V Circuits  E ) P  ->  F ( V Trails  E ) P )
 
Theoremcyclispth 27763 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( V Paths  E ) P )
 
Theoremcycliscrct 27764 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( V Circuits  E ) P )
 
Theoremcyclnspth 27765 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F  =/=  (/)  ->  ( F ( V Cycles  E ) P 
 ->  -.  F ( V SPaths  E ) P ) )
 
Theoremcycliswlk 27766 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
 
Theoremcyclispthon 27767 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( ( P `  0
 ) ( V PathOn  E ) ( P `  0 ) ) P )
 
Theoremfargshiftlem 27768 If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  (
 ( N  e.  NN0  /\  X  e.  ( 0..^ N ) )  ->  ( X  +  1
 )  e.  ( 1
 ... N ) )
 
Theoremfargshiftfv 27769* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  ( X  e.  ( 0..^ N )  ->  ( G `  X )  =  ( F `  ( X  +  1
 ) ) ) )
 
Theoremfargshiftf 27770* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  G : ( 0..^ ( # `  F ) ) --> dom  E )
 
Theoremfargshiftf1 27771* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) -1-1-> dom  E )  ->  G :
 ( 0..^ ( # `  F ) ) -1-1-> dom  E )
 
Theoremfargshiftfo 27772* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) -onto-> dom 
 E )  ->  G : ( 0..^ ( # `  F ) )
 -onto->
 dom  E )
 
Theoremfargshiftfva 27773* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  ( A. k  e.  ( 1 ... N ) ( E `  ( F `  k ) )  =  [_ k  /  x ]_ P  ->  A. l  e.  ( 0..^ N ) ( E `
  ( G `  l ) )  = 
 [_ ( l  +  1 )  /  x ]_ P ) )
 
Theoremeupatrl 27774* An Eulerian path is a trail.

Unfortunately, the edge function  F of an Eulerian path has the domain  ( 1 ... ( # `  F
) ), whereas the edge functions of all kinds of walks defined here have the domain  ( 0..^ ( # `  F
) ) (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 27768, fargshiftfv 27769, etc.). The definition of an Eulerian path and all related theorems should be modified as soon as the graph theory is integrated into the main part of set.mm. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( F ( V EulPaths  E ) P  ->  G ( V Trails  E ) P )
 
Theoremusgrcyclnl1 27775 In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `
  F )  =/=  1 )
 
Theoremusgrcyclnl2 27776 In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `
  F )  =/=  2 )
 
Theorem3cycl3dv 27777 In a simple graph, the vertices of a 3-cycle are mutually different. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A ) )
 
Theoremnvnencycllem 27778 Lemma for 3v3e3cycl1 27779 and 4cycl4v4e 27801. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( ( Fun  E  /\  F  e. Word  dom  E ) 
 /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) ) 
 ->  ( ( E `  ( F `  X ) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
 
Theorem3v3e3cycl1 27779* If there is a cycle of length 3 in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( Fun  E  /\  F ( V Cycles  E ) P  /\  ( # `  F )  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
 
Theoremconstr3lem1 27780 Lemma for constr3trl 27794 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( F  e.  _V 
 /\  P  e.  _V )
 
Theoremconstr3lem2 27781 Lemma for constr3trl 27794 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( # `  F )  =  3
 
Theoremconstr3lem4 27782 Lemma for constr3trl 27794 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( P `  0
 )  =  A  /\  ( P `  1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3
 )  =  A ) ) )
 
Theoremconstr3lem5 27783 Lemma for constr3trl 27794 etc. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( F `
  0 )  =  ( `' E `  { A ,  B }
 )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E ` 
 { C ,  A } ) )
 
Theoremconstr3lem6 27784 Lemma for constr3pthlem3 27792. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A ) ) 
 ->  ( { ( P `
  0 ) ,  ( P `  3
 ) }  i^i  {
 ( P `  1
 ) ,  ( P `
  2 ) }
 )  =  (/) )
 
Theoremconstr3trllem1 27785 Lemma for constr3trl 27794. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  F  e. Word  dom  E )
 
Theoremconstr3trllem2 27786 Lemma for constr3trl 27794. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  Fun  `' F )
 
Theoremconstr3trllem3 27787 Lemma for constr3trl 27794. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P :
 ( 0 ... ( # `
  F ) ) --> V )
 
Theoremconstr3trllem4 27788 Lemma for constr3trl 27794. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P :
 ( 0 ... 3
 ) --> V )
 
Theoremconstr3trllem5 27789* Lemma for constr3trl 27794. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  A. k  e.  (
 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 )
 
Theoremconstr3pthlem1 27790 Lemma for constr3pth 27795. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( P  |`  ( 1..^ ( # `  F ) ) )  =  { <. 1 ,  B >. ,  <. 2 ,  C >. } )
 
Theoremconstr3pthlem2 27791 Lemma for constr3pth 27795. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  B  =/=  C ) 
 ->  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) ) )
 
Theoremconstr3pthlem3 27792 Lemma for constr3pth 27795. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A ) ) 
 ->  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/) )
 
Theoremconstr3cycllem1 27793 Lemma for constr3cycl 27796. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `
  0 )  =  ( P `  ( # `
  F ) ) )
 
Theoremconstr3trl 27794 Construction of a trail from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  F ( V Trails  E ) P )
 
Theoremconstr3pth 27795 Construction of a path from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  F ( V Paths  E ) P )
 
Theoremconstr3cycl 27796 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  ( F ( V Cycles  E ) P  /\  ( # `  F )  =  3 ) )
 
Theoremconstr3cyclp 27797 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  ( F ( V Cycles  E ) P  /\  ( # `  F )  =  3  /\  ( P `  0 )  =  A ) )
 
Theoremconstr3cyclpe 27798* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3  /\  ( p `  0 )  =  A ) )
 
Theorem3v3e3cycl2 27799* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
 |-  ( V USGrph  E  ->  ( E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E )  ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 ) ) )
 
Theorem3v3e3cycl 27800* If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
 |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
 )  =  3 )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  {
 c ,  a }  e.  ran  E ) ) )
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