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Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisusgr 39401* The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e.  U  ->  ( G  e. USGraph  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 ) )
 
Theoremuspgrf 39402* The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USPGraph  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremusgrf 39403* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 )
 
Theoremisusgrs 39404* The property of being a simple graph, simplified version of isusgr 39401. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) (Proof shortened by AV, 24-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e.  U  ->  ( G  e. USGraph  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 ) )
 
Theoremusgrfs 39405* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 39403. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 )
 
Theoremusgrfun 39406 The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.) (Revised by AV, 13-Oct-2020.)
 |-  ( G  e. USGraph  ->  Fun  (iEdg `  G ) )
 
Theoremusgrusgra 39407 A simple graph represented by a class induces a representation as binary relation. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
 |-  ( G  e. USGraph  ->  (Vtx `  G ) USGrph  (iEdg `  G ) )
 
Theoremusgredgss 39408* The set of edges of a simple graph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
 |-  ( G  e. USGraph  ->  (Edg `  G )  C_  { x  e.  ~P (Vtx `  G )  |  ( # `  x )  =  2 }
 )
 
Theoremedgusgr 39409 An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
 |-  (
 ( G  e. USGraph  /\  E  e.  (Edg `  G )
 )  ->  ( E  e.  ~P (Vtx `  G )  /\  ( # `  E )  =  2 )
 )
 
Theoremisusgrop 39410* The property of being an undirected simple graph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 30-Nov-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( <. V ,  E >.  e. USGraph 
 <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 ) )
 
Theoremusgrop 39411 A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.)
 |-  ( G  e. USGraph  ->  <. (Vtx `  G ) ,  (iEdg `  G ) >.  e. USGraph  )
 
Theoremisausgr 39412* The property of an unordered pair to be an alternatively defined simple graph, 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 } ) )
 
Theoremausgrusgrb 39413* The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)
 |-  G  =  { <. v ,  e >.  |  e  C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V G E  <->  <. V ,  (  _I  |`  E ) >.  e. USGraph  ) )
 
Theoremusgrausgri 39414* A simple graph represented by an alternatively defined simple graph. (Contributed by AV, 15-Oct-2020.)
 |-  G  =  { <. v ,  e >.  |  e  C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   =>    |-  ( H  e. USGraph  ->  (Vtx `  H ) G (Edg `  H )
 )
 
Theoremausgrumgri 39415* If an alternatively defined simple graph has the vertices and edges of an arbitrary graph, the arbitrary graph is an undirected multigraph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 25-Nov-2020.)
 |-  G  =  { <. v ,  e >.  |  e  C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   =>    |-  ( ( H  e.  W  /\  (Vtx `  H ) G (Edg `  H )  /\  Fun  (iEdg `  H ) ) 
 ->  H  e. UMGraph  )
 
Theoremausgrusgri 39416* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)
 |-  G  =  { <. v ,  e >.  |  e  C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   &    |-  O  =  { f  |  f : dom  f -1-1-> ran  f }   =>    |-  ( ( H  e.  W  /\  (Vtx `  H ) G (Edg `  H )  /\  (iEdg `  H )  e.  O )  ->  H  e. USGraph  )
 
Theoremusgrausgrb 39417* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
 |-  G  =  { <. v ,  e >.  |  e  C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   &    |-  O  =  { f  |  f : dom  f -1-1-> ran  f }   =>    |-  ( ( H  e.  W  /\  (iEdg `  H )  e.  O )  ->  ( (Vtx `  H ) G (Edg `  H ) 
 <->  H  e. USGraph  ) )
 
Theoremusgredgop 39418 An edge of a 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.) (Revised by AV, 15-Oct-2020.)
 |-  (
 ( G  e. USGraph  /\  E  =  (iEdg `  G )  /\  X  e.  dom  E )  ->  ( ( E `
  X )  =  { M ,  N } 
 <-> 
 <. X ,  { M ,  N } >.  e.  E ) )
 
Theoremusgrf1o 39419 The edge function of a simple graph is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  E : dom  E -1-1-onto-> ran  E )
 
Theoremusgrf1 39420 The edge function of a simple graph is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  E : dom  E -1-1-> ran  E )
 
Theoremuspgrf1oedg 39421 The edge function of a simple graph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USPGraph  ->  E : dom  E -1-1-onto-> (Edg `  G )
 )
 
Theoremusgrss 39422 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 15-Oct-2020.)
 |-  E  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E )  ->  ( E `  X ) 
 C_  V )
 
Theoremuspgrushgr 39423 A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
 |-  ( G  e. USPGraph  ->  G  e. USHGraph  )
 
Theoremuspgrupgr 39424 A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
 |-  ( G  e. USPGraph  ->  G  e. UPGraph  )
 
Theoremuspgrupgrushgr 39425 A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.)
 |-  ( G  e. USPGraph  <->  ( G  e. UPGraph  /\  G  e. USHGraph  ) )
 
Theoremusgruspgr 39426 A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
 |-  ( G  e. USGraph  ->  G  e. USPGraph  )
 
Theoremusgrumgr 39427 A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.)
 |-  ( G  e. USGraph  ->  G  e. UMGraph  )
 
Theoremusgrumgruspgr 39428 A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020.)
 |-  ( G  e. USGraph  <->  ( G  e. UMGraph  /\  G  e. USPGraph  ) )
 
Theoremusgruspgrb 39429* A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)
 |-  ( G  e. USGraph  <->  ( G  e. USPGraph  /\  A. e  e.  (Edg `  G ) ( # `  e )  =  2 ) )
 
Theoremusgrupgr 39430 A simple graph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.)
 |-  ( G  e. USGraph  ->  G  e. UPGraph  )
 
Theoremusgruhgr 39431 A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.)
 |-  ( G  e. USGraph  ->  G  e. UHGraph  )
 
Theoremusgrislfuspgr 39432* A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  <->  ( G  e. USPGraph  /\  I : dom  I --> { x  e.  ~P V  |  2 
 <_  ( # `  x ) } ) )
 
Theoremuspgrun 39433 The union  U of two simple pseudographs  G and  H with the same vertex set  V is a pseudograph with the vertex  V and the union  ( E  u.  F ) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
 |-  ( ph  ->  G  e. USPGraph  )   &    |-  ( ph  ->  H  e. USPGraph  )   &    |-  E  =  (iEdg `  G )   &    |-  F  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  (Vtx `  U )  =  V )   &    |-  ( ph  ->  (iEdg `  U )  =  ( E  u.  F ) )   =>    |-  ( ph  ->  U  e. UPGraph  )
 
Theoremuspgrunop 39434 The union of two simple pseudographs (with the same vertex set): If  <. V ,  E >. and  <. V ,  F >. are simple pseudographs, then  <. V ,  E  u.  F >. is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
 |-  ( ph  ->  G  e. USPGraph  )   &    |-  ( ph  ->  H  e. USPGraph  )   &    |-  E  =  (iEdg `  G )   &    |-  F  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  <. V ,  ( E  u.  F ) >.  e. UPGraph  )
 
Theoremusgrun 39435 The union  U of two simple graphs  G and  H with the same vertex set  V is a multigraph (not necessarily a simple graph!) with the vertex  V and the union  ( E  u.  F
) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
 |-  ( ph  ->  G  e. USGraph  )   &    |-  ( ph  ->  H  e. USGraph  )   &    |-  E  =  (iEdg `  G )   &    |-  F  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  (Vtx `  U )  =  V )   &    |-  ( ph  ->  (iEdg `  U )  =  ( E  u.  F ) )   =>    |-  ( ph  ->  U  e. UMGraph  )
 
Theoremusgrunop 39436 The union of two simple graphs (with the same vertex set): If  <. V ,  E >. and  <. V ,  F >. are simple graphs, then  <. V ,  E  u.  F >. is a multigraph (not necessarily a simple graph!) - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.)
 |-  ( ph  ->  G  e. USGraph  )   &    |-  ( ph  ->  H  e. USGraph  )   &    |-  E  =  (iEdg `  G )   &    |-  F  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  <. V ,  ( E  u.  F ) >.  e. UMGraph  )
 
Theoremusgredg2 39437 The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E )  ->  ( # `  ( E `
  X ) )  =  2 )
 
Theoremusgredg2ALT 39438 Alternate proof of usgredg2 39437, not using umgredg2 39345. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E )  ->  ( # `  ( E `
  X ) )  =  2 )
 
Theoremusgredgprv 39439 In a simple graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E )  ->  ( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V ) ) )
 
TheoremusgredgprvALT 39440 Alternate proof of usgredgprv 39439, using usgredg2 39437 instead of umgredgprv 39352. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  E  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E )  ->  ( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V ) ) )
 
Theoremusgredgappr 39441 An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge). Analogue of usgredg2 39437. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.)
 |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  C  e.  E )  ->  ( # `
  C )  =  2 )
 
Theoremusgrpredgav 39442 An edge of a simple graph always connects two vertices. Analogue of usgredgprv 39439. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
 |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. USGraph  /\  { M ,  N }  e.  E )  ->  ( M  e.  V  /\  N  e.  V )
 )
 
Theoremedgassv2 39443 An edge of a simple graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  C  e.  E )  ->  ( C  C_  V  /\  ( # `
  C )  =  2 ) )
 
Theoremusgredg 39444* 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.) (Revised by AV, 17-Oct-2020.) (Shortened by AV, 25-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  C  e.  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  C  =  { a ,  b } ) )
 
Theoremusgrnloopv 39445 In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  M  e.  W )  ->  (
 ( E `  X )  =  { M ,  N }  ->  M  =/=  N ) )
 
TheoremusgrnloopvALT 39446 Alternate proof of usgrnloopv 39445, not using umgrnloopv 39351. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  M  e.  W )  ->  (
 ( E `  X )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgrnloop 39447* In a simple graph, there is no loop, i.e. 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.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
TheoremusgrnloopALT 39448* Alternate proof of usgrnloop 39447, not using umgrnloop 39353. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgrnloop0 39449* A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
 
Theoremusgrnloop0ALT 39450* Alternate proof of usgrnloop0 39449, not using umgrnloop0 39354. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
 
Theoremusgredgne 39451 An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv 39445 resp. usgrnloop 39447. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
 |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  { M ,  N }  e.  E )  ->  M  =/=  N )
 
Theoremusgrf1oedg 39452 The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.)
 |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. USGraph  ->  I : dom  I -1-1-onto-> E )
 
Theoremuhgr2edg 39453* If a vertex is adjacent to two different vertices in a hypergraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( ( G  e. UHGraph  /\  A  =/=  B ) 
 /\  ( A  e.  V  /\  B  e.  V  /\  N  e.  V ) 
 /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E )
 )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x )  /\  N  e.  ( I `  y ) ) )
 
Theoremumgr2edg 39454* If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( ( G  e. UMGraph  /\  A  =/=  B ) 
 /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E )
 )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x )  /\  N  e.  ( I `  y ) ) )
 
Theoremusgr2edg 39455* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( ( G  e. USGraph  /\  A  =/=  B ) 
 /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E )
 )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x )  /\  N  e.  ( I `  y ) ) )
 
Theoremusgr2edg1 39456* 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.) (Revised by AV, 17-Oct-2020.)
 |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( ( G  e. USGraph  /\  A  =/=  B ) 
 /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E )
 )  ->  -.  E! x  e.  dom  I  N  e.  ( I `  x ) )
 
Theoremusgredg3 39457* The value of the "edge function" of a simple graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E )  ->  E. x  e.  V  E. y  e.  V  ( x  =/=  y  /\  ( E `  X )  =  { x ,  y } ) )
 
Theoremusgredg4 39458* For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E  /\  Y  e.  ( E `  X ) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
 )
 
Theoremusgredgreu 39459* For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  X  e.  dom  E  /\  Y  e.  ( E `  X ) )  ->  E! y  e.  V  ( E `  X )  =  { Y ,  y }
 )
 
Theoremusgredg2vtx 39460* For a vertex incident to an edge there is another vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 5-Dec-2020.)
 |-  (
 ( G  e. USGraph  /\  E  e.  (Edg `  G )  /\  Y  e.  E ) 
 ->  E. y  e.  (Vtx `  G ) E  =  { Y ,  y }
 )
 
Theoremuspgredg2vtxeu 39461* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 6-Dec-2020.)
 |-  (
 ( G  e. USPGraph  /\  E  e.  (Edg `  G )  /\  Y  e.  E ) 
 ->  E! y  e.  (Vtx `  G ) E  =  { Y ,  y }
 )
 
Theoremusgredg2vtxeu 39462* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)
 |-  (
 ( G  e. USGraph  /\  E  e.  (Edg `  G )  /\  Y  e.  E ) 
 ->  E! y  e.  (Vtx `  G ) E  =  { Y ,  y }
 )
 
Theoremusgredg2vtxeuALT 39463* Alternate proof of usgredg2vtxeu 39462, using edgiedgb 39377, the general translation from  (iEdg `  G ) to  (Edg `  G ). (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( G  e. USGraph  /\  E  e.  (Edg `  G )  /\  Y  e.  E ) 
 ->  E! y  e.  (Vtx `  G ) E  =  { Y ,  y }
 )
 
Theoremuspgredg2vlem 39464* Lemma for uspgredg2v 39465. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USPGraph  /\  Y  e.  A )  ->  ( iota_ z  e.  V  Y  =  { N ,  z } )  e.  V )
 
Theoremuspgredg2v 39465* In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( y  e.  A  |->  ( iota_ z  e.  V  y  =  { N ,  z } ) )   =>    |-  ( ( G  e. USPGraph  /\  N  e.  V )  ->  F : A -1-1-> V )
 
Theoremusgredg2vlem1 39466* Lemma 1 for usgredg2v 39468. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   =>    |-  (
 ( G  e. USGraph  /\  Y  e.  A )  ->  ( iota_
 z  e.  V  ( E `  Y )  =  { z ,  N } )  e.  V )
 
Theoremusgredg2vlem2 39467* Lemma 2 for usgredg2v 39468. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   =>    |-  (
 ( G  e. USGraph  /\  Y  e.  A )  ->  ( I  =  ( iota_ z  e.  V  ( E `  Y )  =  {
 z ,  N }
 )  ->  ( E `  Y )  =  { I ,  N }
 ) )
 
Theoremusgredg2v 39468* In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   &    |-  F  =  ( y  e.  A  |->  ( iota_ z  e.  V  ( E `  y )  =  { z ,  N } ) )   =>    |-  ( ( G  e. USGraph  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgredgleord 39469* In a simple 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.) (Revised by AV, 18-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  N  e.  V )  ->  ( # `
  { x  e. 
 dom  E  |  N  e.  ( E `  x ) } )  <_  ( # `
  V ) )
 
Theoremushgredgedga 39470* In a simple hypergraph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 11-Dec-2020.)
 |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   &    |-  A  =  { i  e.  dom  I  |  N  e.  ( I `  i ) }   &    |-  B  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( x  e.  A  |->  ( I `
  x ) )   =>    |-  ( ( G  e. USHGraph  /\  N  e.  V )  ->  F : A -1-1-onto-> B )
 
Theoremusgredgedga 39471* In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
 |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   &    |-  A  =  { i  e.  dom  I  |  N  e.  ( I `  i ) }   &    |-  B  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( x  e.  A  |->  ( I `
  x ) )   =>    |-  ( ( G  e. USGraph  /\  N  e.  V ) 
 ->  F : A -1-1-onto-> B )
 
Theoremushgredgedgaloop 39472* In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex and the set of loops at this vertex. (Contributed by AV, 11-Dec-2020.)
 |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   &    |-  A  =  { i  e.  dom  I  |  ( I `  i )  =  { N } }   &    |-  B  =  {
 e  e.  E  |  e  =  { N } }   &    |-  F  =  ( x  e.  A  |->  ( I `  x ) )   =>    |-  ( ( G  e. USHGraph  /\  N  e.  V )  ->  F : A -1-1-onto-> B )
 
Theoremuspgredgaleord 39473* In a simple pseudograph 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.) (Revised by AV, 6-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USPGraph  /\  N  e.  V )  ->  ( # `
  { e  e.  E  |  N  e.  e } )  <_  ( # `
  V ) )
 
Theoremusgredgaleord 39474* In a simple 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.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  N  e.  V )  ->  ( # `
  { e  e.  E  |  N  e.  e } )  <_  ( # `
  V ) )
 
TheoremusgredgaleordALT 39475* In a simple 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.) (Revised by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  N  e.  V )  ->  ( # `
  { e  e.  E  |  N  e.  e } )  <_  ( # `
  V ) )
 
21.33.8.8  Examples for graphs
 
Theoremusgr0e 39476 The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.)
 |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  (iEdg `  G )  =  (/) )   =>    |-  ( ph  ->  G  e. USGraph  )
 
Theoremusgr0vb 39477 The null graph, with no vertices, is a simple graph iff the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 16-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  (/) )  ->  ( G  e. USGraph  <->  (iEdg `  G )  =  (/) ) )
 
Theoremuhgr0v0e 39478 The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  V  =  (/) )  ->  E  =  (/) )
 
Theoremuhgr0vsize0 39479 The size of a hypergraph with 0 vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 7-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  ( # `
  V )  =  0 )  ->  ( # `
  E )  =  0 )
 
Theoremusgr0v 39480 The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  (/)  /\  (iEdg `  G )  =  (/) )  ->  G  e. USGraph  )
 
Theoremuhgr0vusgr 39481 The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.)
 |-  (
 ( G  e. UHGraph  /\  (Vtx `  G )  =  (/) )  ->  G  e. USGraph  )
 
Theoremusgr0 39482 The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
 |-  (/)  e. USGraph
 
Theoremuspgr1e 39483 A simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { B ,  C } >. } )   =>    |-  ( ph  ->  G  e. USPGraph  )
 
Theoremusgr1e 39484 A simple graph with one edge ( with additional assumption that  B  =/=  C since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { B ,  C } >. } )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  G  e. USGraph  )
 
Theoremusgr0eop 39485 The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
 |-  ( V  e.  W  ->  <. V ,  (/) >.  e. USGraph  )
 
Theoremuspgr1eop 39486 A simple pseudograph with (at least) two vertices and one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  <. V ,  { <. A ,  { B ,  C } >. } >.  e. USPGraph  )
 
Theoremuspgr1ewop 39487 A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
 |-  (
 ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  -> 
 <. V ,  <" { A ,  B } "> >.  e. USPGraph  )
 
Theoremuspgr1v1eop 39488 A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.)
 |-  (
 ( V  e.  W  /\  A  e.  X  /\  B  e.  V )  -> 
 <. V ,  { <. A ,  { B } >. } >.  e. USPGraph  )
 
Theoremusgr1eop 39489 A simple graph with (at least) two different vertices and one edge. If the two vertices were not different, the edge would be a loop. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  ( B  =/=  C  ->  <. V ,  { <. A ,  { B ,  C } >. } >.  e. USGraph  ) )
 
Theoremuspgr2v1e2w 39490 A simple pseudograph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
 |-  (
 ( A  e.  X  /\  B  e.  Y ) 
 ->  <. { A ,  B } ,  <" { A ,  B } "> >.  e. USPGraph  )
 
Theoremusgr2v1e2w 39491 A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
 |-  (
 ( A  e.  X  /\  B  e.  Y  /\  A  =/=  B )  ->  <. { A ,  B } ,  <" { A ,  B } "> >.  e. USGraph  )
 
Theoremedg0usgr 39492 A class without edges is a simple graph. Since  ran 
F  =  (/) does not generally imply  Fun  F, but  Fun  (iEdg `  G ) is required for  G to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Edg `  G )  =  (/)  /\  Fun  (iEdg `  G ) )  ->  G  e. USGraph  )
 
Theoremusgr1vr 39493 A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.)
 |-  (
 ( A  e.  X  /\  (Vtx `  G )  =  { A } )  ->  ( G  e. USGraph  ->  (iEdg `  G )  =  (/) ) )
 
Theoremusgr1v 39494 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.) (Revised by AV, 18-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  { A } )  ->  ( G  e. USGraph  <->  (iEdg `  G )  =  (/) ) )
 
Theoremusgr1v0edg 39495 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.) (Revised by AV, 18-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  { A }  /\  Fun  (iEdg `  G )
 )  ->  ( G  e. USGraph  <-> 
 (Edg `  G )  =  (/) ) )
 
Theoremusgrexmpllem 39496 Lemma for usgrexmpl 39499. (Contributed by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  ( (Vtx `  G )  =  V  /\  (iEdg `  G )  =  E )
 
Theoremusgrexmplvtx 39497 The vertices  0 ,  1 ,  2 ,  3 ,  4 of the graph  G  =  <. V ,  E >.. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  (Vtx `  G )  =  ( { 0 ,  1 ,  2 }  u.  { 3 ,  4 } )
 
Theoremusgrexmpledg 39498 The edges  { 0 ,  1 } ,  {
1 ,  2 } ,  { 2 ,  0 } ,  {
0 ,  3 } of the graph  G  =  <. V ,  E >.. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  (Edg `  G )  =  ( { { 0 ,  1 } ,  { 1 ,  2 } }  u.  { { 2 ,  0 } ,  { 0 ,  3 } }
 )
 
Theoremusgrexmpl 39499  G is a simple 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.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  G  e. USGraph
 
Theoremgriedg0prc 39500* The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.)
 |-  U  =  { <. v ,  e >.  |  e : (/) --> (/) }   =>    |-  U  e/  _V
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