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Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremusgranloop 21401* In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.)
 |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgranloop0 21402* A simple undirected graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
 
Theoremusgraedgrn 21403 An edge of an undirected simple graph without loops always connects two different vertices. (Contributed by Alexander van der Vekens, 2-Sep-2017.)
 |-  ( ( V USGrph  E  /\  { M ,  N }  e.  ran  E ) 
 ->  M  =/=  N )
 
Theoremusgra2edg 21404* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
 )  /\  ( { N ,  b }  e.  ran  E  /\  {
 c ,  N }  e.  ran  E ) ) 
 ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `
  x )  /\  N  e.  ( E `  y ) ) )
 
Theoremusgra2edg1 21405* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
 )  /\  ( { N ,  b }  e.  ran  E  /\  {
 c ,  N }  e.  ran  E ) ) 
 ->  -.  E! x  e. 
 dom  E  N  e.  ( E `  x ) )
 
Theoremusgrarnedg 21406* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
 |-  ( ( V USGrph  E  /\  Y  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) )
 
Theoremusgraedg3 21407* The value of the "edge function" of a graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  E. x  e.  V  E. y  e.  V  ( x  =/=  y  /\  ( E `  X )  =  { x ,  y } ) )
 
Theoremusgraedg4 21408* The value of the "edge function" of a graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E  /\  Y  e.  ( E `
  X ) ) 
 ->  E. y  e.  V  ( E `  X )  =  { Y ,  y } )
 
Theoremusgraedgreu 21409* The value of the "edge function" of a graph is a uniquely determined set containing two elements (the endvertices of the corresponding edge). Concretising usgraedg4 21408. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E  /\  Y  e.  ( E `
  X ) ) 
 ->  E! y  e.  V  ( E `  X )  =  { Y ,  y } )
 
Theoremusgrarnedg1 21410* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
 |-  ( ( V USGrph  E  /\  E. y  e.  ran  E  y  =  ( E `
  I ) ) 
 ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  ( E `  I
 )  =  { a ,  b } ) )
 
Theoremusgra1v 21411 A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( { A } USGrph  E  <->  E  =  (/) )
 
Theoremusgraidx2vlem1 21412* Lemma 1 for usgraidx2v 21414. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   =>    |-  ( ( V USGrph  E  /\  Y  e.  A ) 
 ->  ( iota_ z  e.  V ( E `  Y )  =  { z ,  N } )  e.  V )
 
Theoremusgraidx2vlem2 21413* Lemma 2 for usgraidx2v 21414. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   =>    |-  ( ( V USGrph  E  /\  Y  e.  A ) 
 ->  ( I  =  (
 iota_ z  e.  V ( E `  Y )  =  { z ,  N } )  ->  ( E `  Y )  =  { I ,  N } ) )
 
Theoremusgraidx2v 21414* The mapping of indices of edges containing a given vertex into the set of vertices is 1-1. The index is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   &    |-  F  =  ( y  e.  A  |->  (
 iota_ z  e.  V ( E `  y )  =  { z ,  N } ) )   =>    |-  ( ( V USGrph  E  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgraedgleord 21415* In a graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  ( ( V USGrph  E  /\  N  e.  V ) 
 ->  ( # `  { x  e.  dom  E  |  N  e.  ( E `  x ) } )  <_  ( # `
  V ) )
 
14.1.3.2  Undirected simple graphs - examples
 
Theoremusgraexvlem 21416 Lemma for usgraexmpl 21422. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  V  =  ( { 0 ,  1 ,  2 }  u.  { 3 ,  4 } )
 
Theoremusgraex0elv 21417 Lemma 0 for usgraexmpl 21422. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  0  e.  V
 
Theoremusgraex1elv 21418 Lemma 1 for usgraexmpl 21422. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  1  e.  V
 
Theoremusgraex2elv 21419 Lemma 2 for usgraexmpl 21422. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  2  e.  V
 
Theoremusgraex3elv 21420 Lemma 3 for usgraexmpl 21422. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  3  e.  V
 
Theoremusgraexmpldifpr 21421 Lemma for usgraexmpl 21422: all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  ( ( { 0 ,  1 }  =/=  { 1 ,  2 } 
 /\  { 0 ,  1 }  =/=  { 2 ,  0 }  /\  { 0 ,  1 }  =/=  { 0 ,  3 } )  /\  ( { 1 ,  2 }  =/=  { 2 ,  0 }  /\  { 1 ,  2 }  =/=  { 0 ,  3 }  /\  {
 2 ,  0 }  =/=  { 0 ,  3 } ) )
 
Theoremusgraexmpl 21422  <. V ,  E >. is a graph of five vertices  0 ,  1 , 
2 ,  3 ,  4, with edges  { 0 ,  1 } ,  {
1 ,  2 } ,  { 2 ,  0 } ,  {
0 ,  3 }. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   &    |-  E  =  <" { 0 ,  1 }  {
 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   =>    |-  V USGrph  E
 
14.1.3.3  Finite undirected simple graphs
 
Theoremfiusgraedgfi 21423* In a finite graph the number of edges which contain a given vertex is also finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  { x  e.  dom  E  |  N  e.  ( E `  x ) }  e.  Fin )
 
Theoremusgrafisindb0 21424 The size of a finite simple graph with 0 vertices is 0. Used for the base case of the induction in usgrafis 21431. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
 |-  ( ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  ( # `  E )  =  0 )
 
Theoremusgrafisindb1 21425 The size of a finite simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
 |-  ( ( V USGrph  E  /\  ( # `  V )  =  1 )  ->  ( # `  E )  =  0 )
 
Theoremusgrares1 21426* Restricting an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( ( V USGrph  E  /\  N  e.  V ) 
 ->  ( V  \  { N } ) USGrph  F )
 
Theoremusgrafilem1 21427* The domain of the edge function is the union of the arguments/indices of all edges containing a specific vertex and the arguments/indices of all edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |- 
 dom  E  =  ( dom  F  u.  { x  e.  dom  E  |  N  e.  ( E `  x ) } )
 
Theoremusgrafilem2 21428* In a graph with a finite number of vertices, the number of edges is finite if and only if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  ( E  e.  Fin  <->  F  e.  Fin ) )
 
Theoremusgrafisinds 21429* In a graph with a finite number of vertices, the number of edges is finite if the number of edges not containing one of the vertices is finite. Used for the step of the induction in usgrafis 21431. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( Y  e.  NN0  ->  ( ( V USGrph  E  /\  ( # `  V )  =  Y  /\  N  e.  V )  ->  ( F  e.  Fin  ->  E  e.  Fin ) ) )
 
Theoremusgrafisbase 21430 Induction base for usgrafis 21431. Main work is done in usgrafisindb0 21424. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
 |-  ( ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  E  e.  Fin )
 
Theoremusgrafis 21431 A simple undirected graph with a finite number of vertices has also only a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  E  e.  Fin )
 
14.1.4  Neighbors, complete graphs and universal vertices
 
Syntaxcnbgra 21432 Extend class notation with Neighbors (of a vertex in a graph).
 class Neighbors
 
Syntaxccusgra 21433 Extend class notation with complete (undirected simple) graphs.
 class ComplUSGrph
 
Syntaxcuvtx 21434 Extend class notation with the universal vertices (in a graph).
 class UnivVertex
 
Definitiondf-nbgra 21435* Define the class of all Neighbors of a vertex in a graph. The neighbors of a vertex are all vertices which are connected with this vertex by an edge. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.)
 |- Neighbors  =  ( g  e.  _V ,  k  e.  ( 1st `  g )  |->  { n  e.  ( 1st `  g )  |  {
 k ,  n }  e.  ran  ( 2nd `  g
 ) } )
 
Definitiondf-cusgra 21436* Define the class of all complete (undirected simple) graphs. An undirected simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v  A. n  e.  (
 v  \  { k } ) { n ,  k }  e.  ran  e ) }
 
Definitiondf-uvtx 21437* Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |- UnivVertex  =  ( v  e.  _V ,  e  e.  _V  |->  { n  e.  v  | 
 A. k  e.  (
 v  \  { n } ) { k ,  n }  e.  ran  e } )
 
14.1.4.1  Neighbors
 
Theoremnbgraop 21438* The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
 
Theoremnbgraop1 21439* The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  ->  ( Fun  E  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  E. i  e.  dom  E ( E `
  i )  =  { N ,  n } } ) )
 
Theoremnbgrael 21440 The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E ) ) )
 
Theoremnbgranv0 21441 There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( N  e/  V  ->  ( <. V ,  E >. Neighbors  N )  =  (/) )
 
Theoremnbusgra 21442* The set of neighbors of a vertex in a graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Proof shortened by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( V USGrph  E  ->  (
 <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
 
Theoremnbgra0nb 21443* A vertex which is not endpoint of an edge has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  (
 A. x  e.  ran  E  N  e/  x  ->  ( <. V ,  E >. Neighbors  N )  =  (/) ) )
 
Theoremnbgraeledg 21444 A class/vertex is a neighbor of another class/vertex if and only if it is an endpoint of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  { N ,  K }  e.  ran  E ) )
 
Theoremnbgraisvtx 21445 Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K ) 
 ->  N  e.  V ) )
 
Theoremnbgra0edg 21446 In a graph with no edges, every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  (/)  ->  ( <. V ,  (/) >. Neighbors  K )  =  (/) )
 
Theoremnbgrassvt 21447 The neighbors of a node in a graph are a subset of all nodes of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  (
 <. V ,  E >. Neighbors  N )  C_  V )
 
Theoremnbgranself 21448* A node in a graph (without loops!) is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  A. v  e.  V  v 
 e/  ( <. V ,  E >. Neighbors  v ) )
 
Theoremnbgrassovt 21449 The neighbors of a vertex are a subset of the other vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  V  ->  (
 <. V ,  E >. Neighbors  N )  C_  ( V  \  { N } ) ) )
 
Theoremnbgranself2 21450 A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  N 
 e/  ( <. V ,  E >. Neighbors  N ) )
 
Theoremnbgrasym 21451 A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  K  e.  ( <. V ,  E >. Neighbors  N ) ) )
 
Theoremnbgracnvfv 21452 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  ( ( V USGrph  E  /\  N  e.  ( <. V ,  E >. Neighbors  U ) )  ->  ( E `  ( `' E `  { U ,  N }
 ) )  =  { U ,  N }
 )
 
Theoremnbgraf1olem1 21453* Lemma 1 for nbgraf1o 21459. For each neighbor of a vertex there is exacly one index for the edge between the vertex and its neighbor. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  N  =  ( <. V ,  E >. Neighbors  U )   &    |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i
 ) }   &    |-  F  =  ( n  e.  N  |->  (
 iota_ i  e.  I
 ( E `  i
 )  =  { U ,  n } ) )   =>    |-  ( ( ( V USGrph  E  /\  U  e.  V )  /\  M  e.  N )  ->  E! i  e.  I  ( E `  i )  =  { U ,  M }
 )
 
Theoremnbgraf1olem2 21454* Lemma 2 for nbgraf1o 21459. The mapping of neighbors to edge indices is a function. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  N  =  ( <. V ,  E >. Neighbors  U )   &    |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i
 ) }   &    |-  F  =  ( n  e.  N  |->  (
 iota_ i  e.  I
 ( E `  i
 )  =  { U ,  n } ) )   =>    |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  F : N --> I )
 
Theoremnbgraf1olem3 21455* Lemma 3 for nbgraf1o 21459. The restricted iota of an edge is the function value of the converse applied to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  N  =  ( <. V ,  E >. Neighbors  U )   &    |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i
 ) }   &    |-  F  =  ( n  e.  N  |->  (
 iota_ i  e.  I
 ( E `  i
 )  =  { U ,  n } ) )   =>    |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( iota_ i  e.  I
 ( E `  i
 )  =  { U ,  M } )  =  ( `' E `  { U ,  M }
 ) )
 
Theoremnbgraf1olem4 21456* Lemma 4 for nbgraf1o 21459. The mapping of neighbors to edge indices applied on a neighbor is the function value of the converse applied on the edge between the vertex and this neighbor. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  N  =  ( <. V ,  E >. Neighbors  U )   &    |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i
 ) }   &    |-  F  =  ( n  e.  N  |->  (
 iota_ i  e.  I
 ( E `  i
 )  =  { U ,  n } ) )   =>    |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( F `  M )  =  ( `' E `  { U ,  M } ) )
 
Theoremnbgraf1olem5 21457* Lemma 5 for nbgraf1o 21459. The mapping of neighbors to edge indices is a one-to-one onto function. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  N  =  ( <. V ,  E >. Neighbors  U )   &    |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i
 ) }   &    |-  F  =  ( n  e.  N  |->  (
 iota_ i  e.  I
 ( E `  i
 )  =  { U ,  n } ) )   =>    |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  F : N -1-1-onto-> I )
 
Theoremnbgraf1o0 21458* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  N  =  ( <. V ,  E >. Neighbors  U )   &    |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i
 ) }   =>    |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  E. f  f : N -1-1-onto-> I )
 
Theoremnbgraf1o 21459* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  E. f  f : ( <. V ,  E >. Neighbors  U ) -1-1-onto-> { i  e.  dom  E  |  U  e.  ( E `  i ) }
 )
 
Theoremnbusgrafi 21460 The class of neighbors of a vertex in a finite graph is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( ( V USGrph  E  /\  N  e.  V  /\  E  e.  Fin )  ->  ( <. V ,  E >. Neighbors  N )  e.  Fin )
 
Theoremedgusgranbfin 21461* The number of neighbors of a vertex in a graph is finite, if and only if the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  ( ( <. V ,  E >. Neighbors  U )  e.  Fin  <->  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin ) )
 
Theoremnb3graprlem1 21462 Lemma 1 for nb3grapr 21464. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )
 
Theoremnb3graprlem2 21463* Lemma 2 for nb3grapr 21464. (Contributed by Alexander van der Vekens, 17-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  ->  (
 ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v } )
 ( <. V ,  E >. Neighbors  A )  =  {
 v ,  w }
 ) )
 
Theoremnb3grapr 21464* The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  ->  (
 ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y }
 ) ( <. V ,  E >. Neighbors  x )  =  {
 y ,  z }
 ) )
 
Theoremnb3grapr2 21465 The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  ->  (
 ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B } ) ) )
 
Theoremnb3gra2nb 21466 If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C } )  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B } ) ) )
 
14.1.4.2  Complete graphs
 
Theoremiscusgra 21467* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ComplUSGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) { n ,  k }  e.  ran  E ) ) )
 
Theoremiscusgra0 21468* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  {
 k } ) { n ,  k }  e.  ran  E ) )
 
Theoremcusisusgra 21469 A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  V USGrph  E )
 
Theoremcusgrarn 21470* In a complete simple graph, the range of the edge function consists of all the pairs with different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  ( V ComplUSGrph  E  ->  ran  E  =  { x  e.  ~P V  |  ( # `  x )  =  2 }
 )
 
Theoremcusgra0v 21471 A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  (/) ComplUSGrph  (/)
 
Theoremcusgra1v 21472 A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |- 
 { A } ComplUSGrph  (/)
 
Theoremcusgra2v 21473 A graph with two (different) vertices is complete if and only if there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B ) 
 ->  ( { A ,  B } USGrph  E  ->  ( { A ,  B } ComplUSGrph  E  <->  { A ,  B }  e.  ran  E ) ) )
 
Theoremnbcusgra 21474 In a complete (undirected simple) graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N }
 ) )
 
Theoremcusgra3v 21475 A graph with three (different) vertices is complete if and only if there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  V  =  { A ,  B ,  C }   =>    |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  V USGrph  E  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( V ComplUSGrph  E  <->  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )
 
Theoremcusgra3vnbpr 21476* The neighbors of a vertex in a graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  V  =  { A ,  B ,  C }   =>    |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  V USGrph  E  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( V ComplUSGrph  E  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y }
 ) ( <. V ,  E >. Neighbors  x )  =  {
 y ,  z }
 ) )
 
Theoremcusgraexilem1 21477* Lemma 1 for cusgraexi 21479. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  ( # `  x )  =  2 }   =>    |-  ( V  e.  W  ->  (  _I  |`  P )  e.  _V )
 
Theoremcusgraexilem2 21478* Lemma 2 for cusgraexi 21479. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  ( # `  x )  =  2 }   =>    |-  ( V  e.  W  ->  V USGrph  (  _I  |`  P ) )
 
Theoremcusgraexi 21479* For each set the identity function restricted to the set of pairs of elements from the given set is an edge function, so that the given set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  ( # `  x )  =  2 }   =>    |-  ( V  e.  W  ->  V ComplUSGrph  (  _I  |`  P ) )
 
Theoremcusgraexg 21480* For each set there is an edge function so that the set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  ( V  e.  W  ->  E. e  V ComplUSGrph  e )
 
Theoremcusgrasizeindb0 21481 Base case of the induction in cusgrasize 21489. The size of a complete simple graph with 0 vertices is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
 |-  ( ( V ComplUSGrph  E  /\  ( # `  V )  =  0 )  ->  ( # `  E )  =  ( ( # `  V )  _C  2
 ) )
 
Theoremcusgrasizeindb1 21482 Base case of the induction in cusgrasize 21489. The size of a complete simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
 |-  ( ( V ComplUSGrph  E  /\  ( # `  V )  =  1 )  ->  ( # `  E )  =  ( ( # `  V )  _C  2
 ) )
 
Theoremcusgrares 21483* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( V  \  { N } ) ComplUSGrph  F )
 
Theoremcusgrasizeindslem1 21484* Lemma 1 for cusgrasizeinds 21487. The domain of the edge function is the union of the arguments/indices of all edges containing a specific vertex and the arguments/indices of all edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |- 
 dom  E  =  ( dom  F  u.  { x  e.  dom  E  |  N  e.  ( E `  x ) } )
 
Theoremcusgrasizeindslem2 21485* Lemma 2 for cusgrasizeinds 21487. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( dom  F  i^i  { x  e.  dom  E  |  N  e.  ( E `  x ) }
 )  =  (/)
 
Theoremcusgrasizeindslem3 21486* Lemma 3 for cusgrasizeinds 21487. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( ( V ComplUSGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  ( # `
  { x  e. 
 dom  E  |  N  e.  ( E `  x ) } )  =  ( ( # `  V )  -  1 ) )
 
Theoremcusgrasizeinds 21487* Part 1 of induction step in cusgrasize 21489. The size of a complete simple graph with  n vertices is  ( n  -  1 ) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( ( V ComplUSGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  ( # `
  E )  =  ( ( ( # `  V )  -  1
 )  +  ( # `  F ) ) )
 
Theoremcusgrasize2inds 21488* Induction step in cusgrasize 21489. If the size of the complete graph with  n vertices reduced by one vertex is " ( n  -  1 ) choose 2", the size of the complete graph with  n vertices is " n choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( Y  e.  NN0  ->  ( ( V ComplUSGrph  E  /\  ( # `  V )  =  Y  /\  N  e.  V )  ->  (
 ( # `  F )  =  ( ( # `  ( V  \  { N } ) )  _C  2 )  ->  ( # `  E )  =  ( ( # `  V )  _C  2 ) ) ) )
 
Theoremcusgrasize 21489 The size of a finite complete simple graph with  n vertices ( n  e.  NN0) is  ( n  _C  2 ) ("
n choose 2") resp.  ( (
( n  -  1 ) * n )  /  2 ). (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  ( ( V ComplUSGrph  E  /\  V  e.  Fin )  ->  ( # `  E )  =  ( ( # `  V )  _C  2
 ) )
 
Theoremcusgrafilem1 21490* Lemma 1 for cusgrafi 21493. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  E. a  e.  V  (
 a  =/=  N  /\  x  =  { a ,  N } ) }   =>    |-  (
 ( V ComplUSGrph  E  /\  N  e.  V )  ->  P  C_ 
 ran  E )
 
Theoremcusgrafilem2 21491* Lemma 2 for cusgrafi 21493. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  E. a  e.  V  (
 a  =/=  N  /\  x  =  { a ,  N } ) }   &    |-  F  =  ( x  e.  ( V  \  { N }
 )  |->  { x ,  N } )   =>    |-  ( ( V  e.  W  /\  N  e.  V )  ->  F : ( V  \  { N } ) -1-1-onto-> P )
 
Theoremcusgrafilem3 21492* Lemma 2 for cusgrafi 21493. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  E. a  e.  V  (
 a  =/=  N  /\  x  =  { a ,  N } ) }   &    |-  F  =  ( x  e.  ( V  \  { N }
 )  |->  { x ,  N } )   =>    |-  ( ( V  e.  W  /\  N  e.  V )  ->  ( -.  V  e.  Fin  ->  -.  P  e.  Fin ) )
 
Theoremcusgrafi 21493 If the size of a complete simple graph is finite, then also its order is finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  ( ( V ComplUSGrph  E  /\  E  e.  Fin )  ->  V  e.  Fin )
 
Theoremusgrasscusgra 21494* An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  A. e  e.  ran  E E. f  e.  ran  F  e  =  f )
 
Theoremsizeusglecusglem1 21495 Lemma 1 for sizeusglecusg 21497. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  (  _I  |`  ran  E ) : ran  E -1-1-> ran  F )
 
Theoremsizeusglecusglem2 21496 Lemma 2 for sizeusglecusg 21497. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F  /\  F  e.  Fin )  ->  E  e.  Fin )
 
Theoremsizeusglecusg 21497 The size of an undirected simple graph with  n vertices is at most the size of a complete simple graph with  n vertices ( n may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ( # `
  E )  <_  ( # `  F ) )
 
Theoremusgramaxsize 21498 The maximum size of an undirected simple graph with  n vertices ( n  e.  NN0) is  ( ( ( n  - 
1 ) * n )  /  2 ). (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  ( # `  E )  <_  ( ( # `  V )  _C  2
 ) )
 
14.1.4.3  Universal vertices
 
Theoremisuvtx 21499* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  { n } ) { k ,  n }  e.  ran  E } )
 
Theoremuvtxel 21500* An element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N } ) { k ,  N }  e.  ran  E ) ) )
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