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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuhgrass 21201 An edge is a subset of vertices, analogous to umgrass 21214. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( V UHGrph  E  /\  F  e.  dom  E )  ->  ( E `  F )  C_  V )
 
Theoremuhgraeq12d 21202 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F )
 )  ->  ( V UHGrph  E  <->  W UHGrph  F ) )
 
Theoremuhgrares 21203 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 21219. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V UHGrph  E  ->  V UHGrph 
 ( E  |`  A ) )
 
Theoremuhgra0 21204 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 21220. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V  e.  W  ->  V UHGrph  (/) )
 
Theoremuhgra0v 21205 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( (/) UHGrph  E  <->  E  =  (/) )
 
Theoremuhgraun 21206 If  <. V ,  E >. and  <. V ,  F >. are hypergraphs, then  <. V ,  E  u.  F >. is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting in two edges between two vertices), analogous to umgraun 21223. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UHGrph  E )   &    |-  ( ph  ->  V UHGrph  F )   =>    |-  ( ph  ->  V UHGrph 
 ( E  u.  F ) )
 
14.1.2  Undirected multigraphs
 
Syntaxcumg 21207 Extend class notation with undirected multigraphs.
 class UMGrph
 
Definitiondf-umgra 21208* Define the class of all undirected multigraphs. A multigraph is a pair  <. V ,  E >. where  E is a function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
 --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Theoremrelumgra 21209 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- 
 Rel UMGrph
 
Theoremisumgra 21210* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremwrdumgra 21211* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V  e.  W  /\  E  e. Word  X )  ->  ( V UMGrph  E  <->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremumgraf2 21212* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgraf 21213* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A ) 
 ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgrass 21214 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  C_  V )
 
Theoremumgran0 21215 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  =/=  (/) )
 
Theoremumgrale 21216 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( # `  ( E `  F ) ) 
 <_  2 )
 
Theoremumgrafi 21217 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  e.  Fin )
 
Theoremumgraex 21218* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  E. x  e.  V  E. y  e.  V  ( E `  F )  =  { x ,  y } )
 
Theoremumgrares 21219 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  V UMGrph 
 ( E  |`  A ) )
 
Theoremumgra0 21220 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V  e.  W  ->  V UMGrph  (/) )
 
Theoremumgra1 21221 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V )
 )  ->  V UMGrph  { <. A ,  { B ,  C } >. } )
 
Theoremumisuhgra 21222 An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V UMGrph  E  ->  V UHGrph  E )
 
Theoremumgraun 21223 If  <. V ,  E >. and  <. V ,  F >. are graphs, then  <. V ,  E  u.  F >. is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   =>    |-  ( ph  ->  V UMGrph 
 ( E  u.  F ) )
 
14.1.3  Undirected simple graphs
 
14.1.3.1  Undirected simple graphs - basics
 
Syntaxcuslg 21224 Extend class notation with undirected (simple) graphs with loops.
 class USLGrph
 
Syntaxcusg 21225 Extend class notation with undirected (simple) graphs (without loops).
 class USGrph
 
Definitiondf-uslgra 21226* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-usgra 21227* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph  <. V ,  E >. where 
E is an injective (one-to-one) function into subsets of  V of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itsself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
 
Theoremreluslgra 21228 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- 
 Rel USLGrph
 
Theoremrelusgra 21229 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- 
 Rel USGrph
 
Theoremuslgrav 21230 The classes of vertices and edges of an undirected simple graph with loops are sets. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USLGrph  E  ->  ( V  e.  _V  /\  E  e.  _V )
 )
 
Theoremusgrav 21231 The classes of vertices and edges of an undirected simple graph without loops are sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( V USGrph  E  ->  ( V  e.  _V  /\  E  e.  _V )
 )
 
Theoremisuslgra 21232* The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USLGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremisusgra 21233* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 ) )
 
Theoremuslgraf 21234* The edge function of an undirected simple graph with loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremusgraf 21235* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 )
 
Theoremisusgra0 21236* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 ) )
 
Theoremusgraf0 21237* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 )
 
Theoremusgrafun 21238 The edge function of an undirected simple graph without loops is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.)
 |-  ( V USGrph  E  ->  Fun 
 E )
 
Theoremisausgra 21239* The property of an unordered pair to be an alternatively defined undirected simple graph without loops (defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
 |-  G  =  { <. v ,  e >.  |  e 
 C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   =>    |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V G E 
 <->  E  C_  { x  e.  ~P V  |  ( # `  x )  =  2 } ) )
 
Theoremausisusgra 21240* The equivalence of the definitions of an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 28-Aug-2017.)
 |-  G  =  { <. v ,  e >.  |  e 
 C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V G E 
 <->  V USGrph  (  _I  |`  E ) ) )
 
Theoremusgraedgop 21241 An edge of an undirected simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  ( ( E `
  X )  =  { M ,  N } 
 <-> 
 <. X ,  { M ,  N } >.  e.  E ) )
 
Theoremusgraf1o 21242 The edge function of an undirected simple graph without loops is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran  E )
 
Theoremusgraf1 21243 The edge function of an undirected simple graph without loops is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> ran  E )
 
Theoremusgrass 21244 An edge is a subset of vertices, analogous to umgrass 21214. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( ( V USGrph  E  /\  F  e.  dom  E )  ->  ( E `  F )  C_  V )
 
Theoremusgraeq12d 21245 Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F )
 )  ->  ( V USGrph  E  <->  W USGrph  F ) )
 
Theoremuslisumgra 21246 An undirected simple graph with loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  V UMGrph  E )
 
Theoremusisuslgra 21247 An undirected simple graph without loops is an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  ( V USGrph  E  ->  V USLGrph  E )
 
Theoremusisumgra 21248 An undirected simple graph without loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USGrph  E  ->  V UMGrph  E )
 
Theoremusgrares 21249 A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 21219. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  V USGrph 
 ( E  |`  A ) )
 
Theoremusgra0 21250 The empty graph, with vertices but no edges, is a graph, analogous to umgra0 21220. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V  e.  W  ->  V USGrph  (/) )
 
Theoremusgra0v 21251 The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  ( (/) USGrph  E  <->  E  =  (/) )
 
Theoremuslgra1 21252 The graph with one edge, analogous to umgra1 21221. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V )
 )  ->  V USLGrph  { <. A ,  { B ,  C } >. } )
 
Theoremusgra1 21253 The graph with one edge, analogous to umgra1 21221 ( with additional assumption that  B  =/=  C since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  ( ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V )
 )  ->  ( B  =/=  C  ->  V USGrph  { <. A ,  { B ,  C } >. } ) )
 
Theoremuslgraun 21254 If  <. V ,  E >. and  <. V ,  F >. are (simple) graphs (with loops), then  <. V ,  E  u.  F >. is a multigraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices), analogous to umgraun 21223. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V USLGrph  E )   &    |-  ( ph  ->  V USLGrph  F )   =>    |-  ( ph  ->  V UMGrph 
 ( E  u.  F ) )
 
Theoremusgraedg2 21255 The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 21216. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  ( # `  ( E `  X ) )  =  2 )
 
Theoremusgraedgprv 21256 In an undirected graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  ( ( E `
  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V ) ) )
 
Theoremusgraedgrnv 21257 An edge of an undirected simple graph always connects two vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  ( ( V USGrph  E  /\  { M ,  N }  e.  ran  E ) 
 ->  ( M  e.  V  /\  N  e.  V ) )
 
Theoremusgranloop 21258* 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.)
 |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgranloop0 21259* 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 21260 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 21261* 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 21262* 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 21263* 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 21264* 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 21265* 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 21266* 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 21265. (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 21267* 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 21268 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 21269* Lemma 1 for usgraidx2v 21271. (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 21270* Lemma 2 for usgraidx2v 21271. (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 21271* 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 21272* 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 21273 Lemma for usgraexmpl 21279. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  V  =  ( { 0 ,  1 ,  2 }  u.  { 3 ,  4 } )
 
Theoremusgraex0elv 21274 Lemma 0 for usgraexmpl 21279. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  0  e.  V
 
Theoremusgraex1elv 21275 Lemma 1 for usgraexmpl 21279. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  1  e.  V
 
Theoremusgraex2elv 21276 Lemma 2 for usgraexmpl 21279. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  2  e.  V
 
Theoremusgraex3elv 21277 Lemma 3 for usgraexmpl 21279. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  3  e.  V
 
Theoremusgraexmpldifpr 21278 Lemma for usgraexmpl 21279: 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 21279  <. 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 21280* 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 21281 The size of a finite simple graph with 0 vertices is 0. Used for the base case of the induction in usgrafis 21288. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
 |-  ( ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  ( # `  E )  =  0 )
 
Theoremusgrafisindb1 21282 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 21283* 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 21284* 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 21285* 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 21286* 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 21288. (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 21287 Induction base for usgrafis 21288. Main work is done in usgrafisindb0 21281. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
 |-  ( ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  E  e.  Fin )
 
Theoremusgrafis 21288 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 21289 Extend class notation with Neighbors (of a vertex in a graph).
 class Neighbors
 
Syntaxccusgra 21290 Extend class notation with complete (undirected simple) graphs.
 class ComplUSGrph
 
Syntaxcuvtx 21291 Extend class notation with the universal vertices (in a graph).
 class UnivVertex
 
Definitiondf-nbgra 21292* 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 21293* 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 21294* 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 21295* 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 21296* 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 21297 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 21298 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 21299* The set of neighbors of a vertex in a graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.)
 |-  ( V USGrph  E  ->  (
 <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
 
Theoremnbgra0nb 21300* 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 )  =  (/) ) )
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