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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0trlon 21501 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 ) )
 
Theorem2trllemF 21502 Lemma 5 for constr2trl 21552. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( ( ( E `
  I )  =  { X ,  Y }  /\  Y  e.  V )  ->  I  e.  dom  E )
 
Theorem2trllemA 21503 Lemma 1 for constr2trl 21552. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( # `  F )  =  2
 
Theorem2trllemB 21504 Lemma 2 for constr2trl 21552. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( 0..^ ( # `  F ) )  =  { 0 ,  1 }
 
Theorem2trllemH 21505 Lemma 3 for constr2trl 21552. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  B  e.  V )  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `
  J )  =  { B ,  C } ) )  ->  F : ( 0..^ ( # `  F ) ) --> dom  E )
 
Theorem2trllemE 21506 Lemma 4 for constr2trl 21552. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  B  e.  V )  /\  I  =/= 
 J  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
 ) )  ->  F : ( 0..^ ( # `  F ) )
 -1-1-> dom  E )
 
Theorem2wlklemA 21507 Lemma for constr2wlk 21551. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( A  e.  V  ->  ( P `  0 )  =  A )
 
Theorem2wlklemB 21508 Lemma for constr2wlk 21551. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( B  e.  V  ->  ( P `  1 )  =  B )
 
Theorem2wlklemC 21509 Lemma for constr2wlk 21551. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( C  e.  V  ->  ( P `  2 )  =  C )
 
Theorem2trllemD 21510 Lemma 4 for constr2trl 21552. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P  Fn  { 0 ,  1 ,  2 } )
 
Theorem2trllemG 21511 Lemma 7 for constr2trl 21552. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P :
 ( 0 ... 2
 ) --> V )
 
Theoremwlkntrllem1 21512 Lemma 1 for wlkntrl 21515: 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 21513* Lemma 2 for wlkntrl 21515: 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 ) ) }
 
Theoremwlkntrllem3 21514* Lemma 3 for wlkntrl 21515: 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 21515* 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 21516* 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
 ) ) )
 
Theorem2wlklem 21517* Lemma for is2wlk 21518 and 2wlklemA 21507. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( A. k  e. 
 { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  (
 k  +  1 ) ) }  <->  ( ( E `
  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P `  1 ) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
  1 ) ,  ( P `  2
 ) } ) )
 
Theoremis2wlk 21518 Properties of a pair of functions to be a walk of length 2 (in an undirected graph). (Contributed by Alexander van der Vekens, 16-Feb-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
 )  ->  ( ( F ( V Walks  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 ) --> dom  E  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P `  1
 ) }  /\  ( E `  ( F `  1 ) )  =  { ( P `  1 ) ,  ( P `  2 ) }
 ) ) ) )
 
14.1.5.2  Paths and simple paths
 
Theorempths 21519* 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 21520* 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 21521 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 21522 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 21523 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 21524 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 21525 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 21526 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 21527 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 21528* 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 21529 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 21530 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 21531 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 21532 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 21533 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 21534* 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 ) )
 
Theoremspthon 21535* The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )
 )  ->  ( A ( V SPathOn  E ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V SPaths  E ) p ) } )
 
Theoremisspthon 21536 Properties of a pair of functions to be a simple path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V SPathOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V SPaths  E ) P ) ) )
 
Theoremisspthonpth 21537 Properties of a pair of functions to be a simple path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 9-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V SPathOn  E ) B ) P  <->  ( F ( V SPaths  E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
 
Theoremspthonprp 21538 Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( F ( A ( V SPathOn  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 SPaths  E ) P ) ) )
 
Theoremspthonisspth 21539 A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  ( F ( A ( V SPathOn  E ) B ) P  ->  F ( V SPaths  E ) P )
 
Theoremspthonepeq 21540 The endpoints of a simple path between two vertices are equal if and only if the path is of length 0 (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( F ( A ( V SPathOn  E ) B ) P  ->  ( A  =  B  <->  ( # `  F )  =  0 )
 )
 
Theoremconstr1trl 21541 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 21542 Lemma 1 for 1pthon 21544. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  F  =  { <. 0 ,  i >. }   &    |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. }   =>    |- 
 Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) )
 
Theorem1pthonlem2 21543 Lemma 2 for 1pthon 21544. (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 21544 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 21545 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 21546* 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 )
 
Theoremconstr2spthlem1 21547 Lemma 1 for constr2spth 21553. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 ->  Fun  `' P )
 
Theorem2pthlem1 21548 Lemma 1 for constr2pth 21554. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  Fun  `' ( P  |`  ( 1..^ 2 ) ) )
 
Theorem2pthlem2 21549 Lemma 2 for constr2pth 21554. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 18-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  B  =/=  C ) )  ->  ( ( P " { 0 ,  2 } )  i^i  ( P " (
 1..^ 2 ) ) )  =  (/) )
 
Theorem2wlklem1 21550* Lemma 1 for constr2wlk 21551. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  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 )
 )  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
 ) )  ->  A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  (
 k  +  1 ) ) } )
 
Theoremconstr2wlk 21551 Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  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 )
 )  ->  ( (
 ( E `  I
 )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } )  ->  F ( V Walks  E ) P ) )
 
Theoremconstr2trl 21552 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  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 )
 )  ->  ( ( I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
 )  ->  F ( V Trails  E ) P ) )
 
Theoremconstr2spth 21553 A simple path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  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 SPaths  E ) P ) )
 
Theoremconstr2pth 21554 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  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 21555 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 21556 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 21557* 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 21558 Lemma for redwlk 21559. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  ( ( F ( V Walks  E ) P 
 /\  1  <_  ( # `
  F ) ) 
 ->  ( # `  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) )  =  ( ( # `  F )  -  1 ) )
 
Theoremredwlk 21559 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 21560* Lemma for wlkdvspth 21561. (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 21561 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 )
 
14.1.5.3  Circuits and cycles
 
Theoremcrcts 21562* 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 21563* 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 21564 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 21565 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 21566 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 21567 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 21568 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 21569 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 21570 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 21571 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 21572 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 21573 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 21574 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 21575* 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 21576* 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 21577* 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 21578* 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 21579* 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 ) )
 
Theoremusgrcyclnl1 21580 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 21581 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 21582 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 21583 Lemma for 3v3e3cycl1 21584 and 4cycl4v4e 21606. (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 21584* 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 21585 Lemma for constr3trl 21599 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 21586 Lemma for constr3trl 21599 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 21587 Lemma for constr3trl 21599 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 21588 Lemma for constr3trl 21599 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 21589 Lemma for constr3pthlem3 21597. (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 21590 Lemma for constr3trl 21599. (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 21591 Lemma for constr3trl 21599. (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 21592 Lemma for constr3trl 21599. (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 21593 Lemma for constr3trl 21599. (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 21594* Lemma for constr3trl 21599. (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 21595 Lemma for constr3pth 21600. (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 21596 Lemma for constr3pth 21600. (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 21597 Lemma for constr3pth 21600. (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 21598 Lemma for constr3cycl 21601. (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 21599 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 21600 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 )
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