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Theorem List for Metamath Proof Explorer - 39201-39300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnbgrsym 39201 A vertex in a graph is a neighbor of a second vertex iff the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.)
 |-  ( G  e.  W  ->  ( N  e.  ( G NeighbVtx  K )  <->  K  e.  ( G NeighbVtx  N ) ) )
 
Theoremnbumgrres 39202* The neighborhood of a vertex in a restricted multigraph (not necessarily valid for a hypergraph, because  N,  K and  M could be connected by one edge, so  M is a neighbor of  K in the original graph, but not in the restricted graph, because the edge between  M and  K, also incident with  N, was removed). (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  ( ( ( G  e. UMGraph  /\  N  e.  V )  /\  K  e.  ( V  \  { N }
 )  /\  M  e.  ( V  \  { N ,  K } ) ) 
 ->  ( M  e.  ( G NeighbVtx  K )  ->  M  e.  ( S NeighbVtx  K )
 ) )
 
Theoremusgrnbcnvfv 39203 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair in a simple graph. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 27-Oct-2020.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e. USGraph  /\  N  e.  ( G NeighbVtx  K )
 )  ->  ( I `  ( `' I `  { K ,  N }
 ) )  =  { K ,  N }
 )
 
Theoremnbusgredgeu 39204* For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.)
 |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  M  e.  ( G NeighbVtx  N )
 )  ->  E! e  e.  E  e  =  { M ,  N }
 )
 
Theoremedgnbusgreu 39205* For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  N  =  ( G NeighbVtx  M )   =>    |-  (
 ( ( G  e. USGraph  /\  M  e.  V ) 
 /\  ( C  e.  E  /\  M  e.  C ) )  ->  E! n  e.  N  C  =  { M ,  n }
 )
 
Theoremnbusgredgeu0 39206* For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  N  =  ( G NeighbVtx  U )   &    |-  I  =  { e  e.  E  |  U  e.  e }   =>    |-  ( ( ( G  e. USGraph  /\  U  e.  V )  /\  M  e.  N )  ->  E! i  e.  I  i  =  { U ,  M }
 )
 
Theoremnbusgrf1o0 39207* The mapping of neighbors of a vertex to edges incident to the vertex is a bijection ( 1-1 onto function) in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  N  =  ( G NeighbVtx  U )   &    |-  I  =  { e  e.  E  |  U  e.  e }   &    |-  F  =  ( n  e.  N  |->  { U ,  n } )   =>    |-  ( ( G  e. USGraph  /\  U  e.  V )  ->  F : N -1-1-onto-> I
 )
 
Theoremnbusgrf1o1 39208* The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  N  =  ( G NeighbVtx  U )   &    |-  I  =  { e  e.  E  |  U  e.  e }   =>    |-  ( ( G  e. USGraph  /\  U  e.  V ) 
 ->  E. f  f : N -1-1-onto-> I )
 
Theoremnbusgrf1o 39209* The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  E. f  f : ( G NeighbVtx  U ) -1-1-onto-> { e  e.  E  |  U  e.  e }
 )
 
Theoremedgusgrnbfin 39210* The number of neighbors of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  (
 ( G NeighbVtx  U )  e. 
 Fin 
 <->  { e  e.  E  |  U  e.  e }  e.  Fin ) )
 
Theoremnbusgrfi 39211 The class of neighbors of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  E  e.  Fin  /\  U  e.  V )  ->  ( G NeighbVtx  U )  e.  Fin )
 
Theoremnbfiusgrfi 39212 The class of neighbors of a vertex in a finite simple graph is a finite set. (Contributed by Alexander van der Vekens, 7-Mar-2018.) (Revised by AV, 28-Oct-2020.)
 |-  (
 ( G  e. FinUSGraph  /\  N  e.  (Vtx `  G )
 )  ->  ( G NeighbVtx  N )  e.  Fin )
 
Theoremnb3grprlem1 39213 Lemma 1 for nb3grapr 25179. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  ( ph  ->  G  e. USGraph  )   &    |-  ( ph  ->  V  =  { A ,  B ,  C } )   &    |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
 )   =>    |-  ( ph  ->  (
 ( G NeighbVtx  A )  =  { B ,  C } 
 <->  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E )
 ) )
 
Theoremnb3grprlem2 39214* Lemma 2 for nb3grapr 25179. (Contributed by Alexander van der Vekens, 17-Oct-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  ( ph  ->  G  e. USGraph  )   &    |-  ( ph  ->  V  =  { A ,  B ,  C } )   &    |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )   =>    |-  ( ph  ->  (
 ( G NeighbVtx  A )  =  { B ,  C } 
 <-> 
 E. v  e.  V  E. w  e.  ( V  \  { v }
 ) ( G NeighbVtx  A )  =  { v ,  w } ) )
 
Theoremnb3grpr 39215* The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  ( ph  ->  G  e. USGraph  )   &    |-  ( ph  ->  V  =  { A ,  B ,  C } )   &    |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )   =>    |-  ( ph  ->  (
 ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E  /\  { C ,  A }  e.  E )  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y }
 ) ( G NeighbVtx  x )  =  { y ,  z } ) )
 
Theoremnb3grpr2 39216 The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  ( ph  ->  G  e. USGraph  )   &    |-  ( ph  ->  V  =  { A ,  B ,  C } )   &    |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )   =>    |-  ( ph  ->  (
 ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E  /\  { C ,  A }  e.  E )  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }  /\  ( G NeighbVtx  C )  =  { A ,  B } ) ) )
 
Theoremnb3gr2nb 39217 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.) (Revised by AV, 28-Oct-2020.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  ( (
 ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C } )  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }  /\  ( G NeighbVtx  C )  =  { A ,  B } ) ) )
 
Theoremuvtxaval 39218* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  (UnivVtx `  G )  =  {
 v  e.  V  |  A. n  e.  ( V  \  { v }
 ) n  e.  ( G NeighbVtx  v ) } )
 
Theoremuvtxael 39219* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( N  e.  (UnivVtx `  G ) 
 <->  ( N  e.  V  /\  A. n  e.  ( V  \  { N }
 ) n  e.  ( G NeighbVtx  N ) ) ) )
 
Theoremuvtxaisvtx 39220 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e.  (UnivVtx `  G )  ->  N  e.  V )
 
Theoremvtxnbuvtx 39221* A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e.  (UnivVtx `  G )  ->  A. n  e.  ( V  \  { N }
 ) n  e.  ( G NeighbVtx  N ) )
 
Theoremuvtxanbgr 39222 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e.  (UnivVtx `  G )  ->  ( V  \  { N } )  C_  ( G NeighbVtx  N ) )
 
Theoremuvtxanbgrvtx 39223* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e.  (UnivVtx `  G )  ->  A. v  e.  ( V  \  { N }
 ) N  e.  ( G NeighbVtx  v ) )
 
Theoremuvtxa0 39224 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( V  =  (/)  ->  (UnivVtx `  G )  =  (/) )
 
Theoremisuvtxa 39225* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e.  W  ->  (UnivVtx `  G )  =  {
 v  e.  V  |  A. k  e.  ( V  \  { v }
 ) E. e  e.  E  { k ,  v }  C_  e } )
 
Theoremuvtxael1 39226* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e.  W  ->  ( N  e.  (UnivVtx `  G ) 
 <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N }
 ) E. e  e.  E  { k ,  N }  C_  e
 ) ) )
 
Theoremuvtxa01vtx0 39227 If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e.  W  /\  E  =  (/) )  ->  ( (UnivVtx `  G )  =/= 
 (/) 
 <->  ( # `  V )  =  1 )
 )
 
Theoremuvtxa01vtx 39228 If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( E  =  (/)  ->  (
 (UnivVtx `  G )  =/=  (/) 
 <->  ( # `  V )  =  1 )
 )
 
Theoremuvtx2vtx1edg 39229* If a graph has two vertices, and there is an edge between the vertices, then each vertex is universal. (Contributed by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e.  W  /\  ( # `  V )  =  2  /\  V  e.  E )  ->  A. v  e.  V  v  e.  (UnivVtx `  G ) )
 
Theoremuvtx2vtx1edgb 39230* If a hypergraph has two vertices, there is an edge between the vertices iff each vertex is universal. (Contributed by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  ( # `
  V )  =  2 )  ->  ( V  e.  E  <->  A. v  e.  V  v  e.  (UnivVtx `  G ) ) )
 
Theoremuvtxnbgr 39231 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  N  e.  (UnivVtx `  G ) )  ->  ( G NeighbVtx  N )  =  ( V  \  { N }
 ) )
 
Theoremuvtxnbgrb 39232 A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  N  e.  V ) 
 ->  ( N  e.  (UnivVtx `  G )  <->  ( G NeighbVtx  N )  =  ( V  \  { N } ) ) )
 
Theoremuvtxusgr 39233* The set of all universal vertices of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. USGraph  ->  (UnivVtx `  G )  =  { n  e.  V  |  A. k  e.  ( V  \  { n } ) { k ,  n }  e.  E } )
 
Theoremuvtxusgrel 39234* A universal vertex, i.e. an element of the set of all universal vertices, of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. USGraph  ->  ( N  e.  (UnivVtx `  G ) 
 <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N }
 ) { k ,  N }  e.  E ) ) )
 
Theoremuvtxanm1nbgr 39235 A universal vertex has  n  -  1 neighbors in a graph with  n vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. FinUSGraph  /\  N  e.  (UnivVtx `  G )
 )  ->  ( # `  ( G NeighbVtx  N ) )  =  ( ( # `  V )  -  1 ) )
 
Theoremnbumgruvtxres 39236* The neighborhood of a universal vertex in a restricted multigraph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  ( ( ( G  e. UMGraph  /\  N  e.  V )  /\  K  e.  ( V  \  { N }
 ) )  ->  (
 ( G NeighbVtx  K )  =  ( V  \  { K } )  ->  ( S NeighbVtx  K )  =  ( V  \  { N ,  K } ) ) )
 
Theoremuvtxumgrres 39237* A universal vertex is universal in a restricted multigraph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  ( ( ( G  e. UMGraph  /\  N  e.  V )  /\  K  e.  ( V  \  { N }
 ) )  ->  ( K  e.  (UnivVtx `  G )  ->  K  e.  (UnivVtx `  S ) ) )
 
Theoremiscplgr 39238* The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( G  e. ComplGraph  <->  A. v  e.  V  v  e.  (UnivVtx `  G ) ) )
 
Theoremcplgruvtxb 39239 An graph is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( G  e. ComplGraph  <->  (UnivVtx `  G )  =  V ) )
 
Theoremiscplgrnb 39240* A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( G  e. ComplGraph  <->  A. v  e.  V  A. n  e.  ( V 
 \  { v }
 ) n  e.  ( G NeighbVtx  v ) ) )
 
Theoremiscplgredg 39241* A graph is complete iff all vertices are connected with each other by (at least) one edge. (Contributed by AV, 10-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e.  W  ->  ( G  e. ComplGraph  <->  A. v  e.  V  A. n  e.  ( V 
 \  { v }
 ) E. e  e.  E  { v ,  n }  C_  e
 ) )
 
Theoremiscusgr 39242 The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
 |-  ( G  e. ComplUSGraph  <->  ( G  e. USGraph  /\  G  e. ComplGraph ) )
 
Theoremcusgrusgr 39243 A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
 |-  ( G  e. ComplUSGraph  ->  G  e. USGraph  )
 
Theoremcusgrcplgr 39244 A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)
 |-  ( G  e. ComplUSGraph  ->  G  e. ComplGraph )
 
Theoremiscusgrvtx 39245* A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. ComplUSGraph  <->  ( G  e. USGraph  /\ 
 A. v  e.  V  v  e.  (UnivVtx `  G ) ) )
 
Theoremcusgruvtxb 39246 A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. USGraph  ->  ( G  e. ComplUSGraph  <-> 
 (UnivVtx `  G )  =  V ) )
 
Theoremiscusgredg 39247* A simple graph is complete iff all vertices are connected by an edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. ComplUSGraph  <->  ( G  e. USGraph  /\ 
 A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) { n ,  k }  e.  E ) )
 
Theoremcusgredg 39248* In a complete simple graph, the edges are all the pairs of different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. ComplUSGraph  ->  E  =  { x  e.  ~P V  |  ( # `  x )  =  2 }
 )
 
Theoremcplgr0 39249 The null graph (with no vertices and no edges) represented by the empty set is a complete graph. (Contributed by AV, 1-Nov-2020.)
 |-  (/)  e. ComplGraph
 
Theoremcusgr0 39250 The null graph (with no vertices and no edges) represented by the empty set is a complete simple graph. (Contributed by AV, 1-Nov-2020.)
 |-  (/)  e. ComplUSGraph
 
Theoremcplgr0v 39251 A graph with no vertices (and therefore no edges) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  V  =  (/) )  ->  G  e. ComplGraph )
 
Theoremcusgr0v 39252 A graph with no vertices (and therefore no edges) is a complete simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  V  =  (/)  /\  (iEdg `  G )  =  (/) )  ->  G  e. ComplUSGraph )
 
Theoremcplgr1v 39253 A graph with one vertex is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  ( # `  V )  =  1 )  ->  G  e. ComplGraph )
 
Theoremcusgr1v 39254 A graph with one vertex and no edges is a complete simple graph. (Contributed by AV, 1-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  ( # `  V )  =  1  /\  (iEdg `  G )  =  (/) )  ->  G  e. ComplUSGraph )
 
Theoremcplgr2v 39255 An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  ( # `
  V )  =  2 )  ->  ( G  e. ComplGraph  <->  V  e.  E ) )
 
Theoremcplgr2vpr 39256 An undirected hypergraph with two (different) vertices is complete iff 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.) (Revised by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  A  =/=  B ) 
 /\  ( G  e. UHGraph  /\  V  =  { A ,  B } ) ) 
 ->  ( G  e. ComplGraph  <->  { A ,  B }  e.  E )
 )
 
Theoremnbcplgr 39257 In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. ComplGraph  /\  N  e.  V )  ->  ( G NeighbVtx  N )  =  ( V  \  { N } ) )
 
Theoremcplgr3v 39258 A multigraph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.)
 |-  E  =  (Edg `  G )   &    |-  (Vtx `  G )  =  { A ,  B ,  C }   =>    |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  G  e. UMGraph  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 ->  ( G  e. ComplGraph  <->  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E  /\  { C ,  A }  e.  E ) ) )
 
Theoremcusgr3vnbpr 39259* The neighbors of a vertex in a simple 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.) (Revised by AV, 5-Nov-2020.)
 |-  E  =  (Edg `  G )   &    |-  (Vtx `  G )  =  { A ,  B ,  C }   &    |-  V  =  (Vtx `  G )   =>    |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  G  e. USGraph  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 ->  ( G  e. ComplGraph  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y }
 ) ( G NeighbVtx  x )  =  { y ,  z } ) )
 
Theoremcplgrop 39260 A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
 |-  ( G  e. ComplGraph  ->  <. (Vtx `  G ) ,  (iEdg `  G ) >.  e. ComplGraph )
 
Theoremcusgrop 39261 A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
 |-  ( G  e. ComplUSGraph  ->  <. (Vtx `  G ) ,  (iEdg `  G ) >.  e. ComplUSGraph )
 
Theoremusgrexi 39262* An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
 |-  P  =  { x  e.  ~P V  |  ( # `  x )  =  2 }   =>    |-  ( V  e.  W  ->  <. V ,  (  _I  |`  P ) >.  e. USGraph  )
 
Theoremcusgrexi 39263* An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
 |-  P  =  { x  e.  ~P V  |  ( # `  x )  =  2 }   =>    |-  ( V  e.  W  ->  <. V ,  (  _I  |`  P ) >.  e. ComplUSGraph )
 
Theoremcusgrexg 39264* For each set there is a set of edges so that the set together with these edges is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
 |-  ( V  e.  W  ->  E. e <. V ,  e >.  e. ComplUSGraph )
 
Theoremcusgrres 39265* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  ( ( G  e. ComplUSGraph  /\  N  e.  V )  ->  S  e. ComplUSGraph )
 
Theoremcusgrsizeindb0 39266 Base case of the induction in cusgrasize 25204. The size of a complete simple graph with 0 vertices, actually of every null graph, is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  ( # `
  V )  =  0 )  ->  ( # `
  E )  =  ( ( # `  V )  _C  2 ) )
 
Theoremcusgrsizeindb1 39267 Base case of the induction in cusgrasize 25204. 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.) (Revised by AV, 7-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  ( # `
  V )  =  1 )  ->  ( # `
  E )  =  ( ( # `  V )  _C  2 ) )
 
Theoremcusgrsizeindslem 39268* Lemma for cusgrsizeinds 39269. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. ComplUSGraph  /\  V  e.  Fin  /\  N  e.  V )  ->  ( # ` 
 { e  e.  E  |  N  e.  e } )  =  (
 ( # `  V )  -  1 ) )
 
Theoremcusgrsizeinds 39269* Part 1 of induction step in cusgrsize 39271. 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.) (Revised by AV, 9-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  (
 ( G  e. ComplUSGraph  /\  V  e.  Fin  /\  N  e.  V )  ->  ( # `  E )  =  ( ( ( # `  V )  -  1 )  +  ( # `  F ) ) )
 
Theoremcusgrsize2inds 39270* Induction step in cusgrasize 25204. 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.) (Revised by AV, 9-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( Y  e.  NN0  ->  (
 ( G  e. ComplUSGraph  /\  ( # `
  V )  =  Y  /\  N  e.  V )  ->  ( ( # `  F )  =  ( ( # `  ( V  \  { N }
 ) )  _C  2
 )  ->  ( # `  E )  =  ( ( # `
  V )  _C  2 ) ) ) )
 
Theoremcusgrsize 39271 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 ), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. ComplUSGraph  /\  V  e.  Fin )  ->  ( # `
  E )  =  ( ( # `  V )  _C  2 ) )
 
Theoremcusgrfilem1 39272* Lemma 1 for cusgrfi 39275. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/= 
 N  /\  x  =  { a ,  N } ) }   =>    |-  ( ( G  e. ComplUSGraph  /\  N  e.  V )  ->  P  C_  (Edg `  G ) )
 
Theoremcusgrfilem2 39273* Lemma 2 for cusgrfi 39275. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/= 
 N  /\  x  =  { a ,  N } ) }   &    |-  F  =  ( x  e.  ( V  \  { N }
 )  |->  { x ,  N } )   =>    |-  ( N  e.  V  ->  F : ( V 
 \  { N }
 )
 -1-1-onto-> P )
 
Theoremcusgrfilem3 39274* Lemma 3 for cusgrfi 39275. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/= 
 N  /\  x  =  { a ,  N } ) }   &    |-  F  =  ( x  e.  ( V  \  { N }
 )  |->  { x ,  N } )   =>    |-  ( N  e.  V  ->  ( V  e.  Fin  <->  P  e.  Fin ) )
 
Theoremcusgrfi 39275 If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. ComplUSGraph  /\  E  e.  Fin )  ->  V  e.  Fin )
 
Theoremusgredgsscusgredg 39276 A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  H )   &    |-  F  =  (Edg `  H )   =>    |-  (
 ( G  e. USGraph  /\  H  e. ComplUSGraph )  ->  E  C_  F )
 
Theoremusgrsscusgr 39277* A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  H )   &    |-  F  =  (Edg `  H )   =>    |-  (
 ( G  e. USGraph  /\  H  e. ComplUSGraph )  ->  A. e  e.  E  E. f  e.  F  e  =  f )
 
Theoremsizusglecusglem1 39278 Lemma 1 for sizusglecusg 39280. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 13-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  H )   &    |-  F  =  (Edg `  H )   =>    |-  (
 ( G  e. USGraph  /\  H  e. ComplUSGraph )  ->  (  _I  |`  E ) : E -1-1-> F )
 
Theoremsizusglecusglem2 39279 Lemma 2 for sizusglecusg 39280. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  H )   &    |-  F  =  (Edg `  H )   =>    |-  (
 ( G  e. USGraph  /\  H  e. ComplUSGraph  /\  F  e.  Fin )  ->  E  e.  Fin )
 
Theoremsizusglecusg 39280 The size of a 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.) (Revised by AV, 13-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  H )   &    |-  F  =  (Edg `  H )   =>    |-  (
 ( G  e. USGraph  /\  H  e. ComplUSGraph )  ->  ( # `  E )  <_  ( # `  F ) )
 
Theoremfusgrmaxsize 39281 The maximum size of a finite simple graph with  n vertices is  ( (
( n  -  1 ) * n )  /  2 ). See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. FinUSGraph  ->  ( # `  E )  <_  ( ( # `  V )  _C  2
 ) )
 
21.33.9  Graph theory (old)
 
21.33.9.1  Undirected hypergraphs
 
Theoremuhgraedgrnv 39282 An edge of an undirected hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  (
 ( V UHGrph  E  /\  F  e.  ran  E  /\  N  e.  F )  ->  N  e.  V )
 
21.33.9.2  Walks, Paths and Cycles
 
Theoremwlkc 39283* A walk as class. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
 |-  ( W  e.  ( V Walks  E )  ->  E. f E. p  f ( V Walks  E ) p )
 
Theoremusgra2pthspth 39284 In a undirected simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  (
 ( V USGrph  E  /\  ( # `  F )  =  2 )  ->  ( F ( V Paths  E ) P  <->  F ( V SPaths  E ) P ) )
 
Theoremspthdifv 39285 The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( F ( V SPaths  E ) P  ->  P :
 ( 0 ... ( # `
  F ) )
 -1-1-> V )
 
Theoremusgra2pthlem1 39286* Lemma for usgra2pth 39287. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  (
 ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P : ( 0
 ... ( # `  F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }
 )  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2 )  ->  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y }
 ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  y  /\  ( P `
  2 )  =  z )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  y }  /\  ( E `
  ( F `  1 ) )  =  { y ,  z } ) ) ) )
 
Theoremusgra2pth 39287* In a undirected simply graph, there is a path of length 2 if and only if there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 )
 -1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `  0
 )  =  x  /\  ( P `  1 )  =  y  /\  ( P `  2 )  =  z )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  y }  /\  ( E `
  ( F `  1 ) )  =  { y ,  z } ) ) ) ) )
 
Theoremusgra2pth0 39288* In a undirected simply graph, there is a path of length 2 if and only if there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 )
 -1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `  0
 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `
  ( F `  1 ) )  =  { z ,  y } ) ) ) ) )
 
Theoremusgra2adedglem1 39289 In an undirected simple graph, two adjacent edges are an unordered pair of unordered pairs. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. }   &    |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 ->  { { A ,  B } ,  { B ,  C } }  =  ( E " ran  F ) ) )
 
21.33.9.3  Vertex degree (extension)
 
Theoremvdusgravaledg 39290* The value of the vertex degree function for simple undirected graphs in terms of edges. (Contributed by Alexander van der Vekens, 9-Jul-2018.)
 |-  (
 ( V USGrph  E  /\  U  e.  V )  ->  ( ( V VDeg  E ) `  U )  =  ( # `  { x  e.  V  |  { U ,  x }  e.  ran  E } ) )
 
Theoremusgrauvtxvd 39291 In a finite complete undirected simple graph with n vertices every vertex has degree (n-1). (Contributed by Alexander van der Vekens, 9-Jul-2018.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin )  ->  ( K  e.  ( V UnivVertex  E )  ->  (
 ( V VDeg  E ) `  K )  =  ( ( # `  V )  -  1 ) ) )
 
Theoremvdcusgra 39292* In a finite complete undirected simple graph with n vertices every vertex has degree (n-1). (Contributed by Alexander van der Vekens, 9-Jul-2018.)
 |-  (
 ( V ComplUSGrph  E  /\  V  e.  Fin )  ->  A. v  e.  V  ( ( V VDeg 
 E ) `  v
 )  =  ( ( # `  V )  -  1 ) )
 
21.33.9.4  Undirected hypergraphs as extensible structures
 
Syntaxcuhgraltv 39293 Extend class notation with undirected hypergraphs as extensible structures.
 class UHGraphALTV
 
Syntaxcushgraltv 39294 Extend class notation with undirected simple hypergraphs as extensible structures.
 class USHGraphALTV
 
Definitiondf-uhgrALTV 39295* Define the class of all undirected hypergraphs. An undirected hypergraph is a set, regarded as set of "vertices", and a function into the powerset of this set (the empty set excluded), regarded as indexed "edges" connecting vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |- UHGraphALTV  =  {
 g  |  [. ( Base `  g )  /  v ]. [. (.ef `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/)
 } ) }
 
Definitiondf-ushgrALTV 39296* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V, representing the (one or more) vertices incident to the edge. This definition corresponds to definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subsets of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E a (non-empty) subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.)
 |- USHGraphALTV  =  {
 g  |  [. ( Base `  g )  /  v ]. [. (.ef `  g )  /  e ]. e : dom  e -1-1-> ( ~P v  \  { (/)
 } ) }
 
TheoremisuhgrALTV 39297 The predicate "is an undirected hypergraph." (Contributed by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  (.ef `  G )   =>    |-  ( G  e.  U  ->  ( G  e. UHGraphALTV  <->  E : dom  E --> ( ~P V  \  { (/)
 } ) ) )
 
TheoremisushgrALTV 39298 The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  (.ef `  G )   =>    |-  ( G  e.  U  ->  ( G  e. USHGraphALTV  <->  E : dom  E -1-1-> ( ~P V  \  { (/)
 } ) ) )
 
TheoremuhgfALTV 39299 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  (.ef `  G )   =>    |-  ( G  e. UHGraphALTV  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
 
TheoremuhgssALTV 39300 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  (.ef `  G )   =>    |-  (
 ( G  e. UHGraphALTV  /\  F  e.  dom  E )  ->  ( E `  F ) 
 C_  V )
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