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Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlpvtx 39301 The endpoints of a loop (which is an edge at index  J) are two (identical) vertices  A. (Contributed by AV, 1-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  J  e.  dom  I  /\  ( I `  J )  =  { A } )  ->  A  e.  (Vtx `  G ) )
 
Theoremushgruhgr 39302 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
 |-  ( G  e. USHGraph  ->  G  e. UHGraph  )
 
Theoremuhgruhgra 39303 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
 |-  (
 ( G  e. UHGraph  /\  V  =  (Vtx `  G )  /\  E  =  (iEdg `  G ) )  ->  V UHGrph  E )
 
Theoremuhgrauhgr 39304 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
 |-  (
 ( V UHGrph  E  /\  V  =  (Vtx `  G )  /\  E  =  (iEdg `  G ) )  ->  ( G  e.  W  ->  G  e. UHGraph  ) )
 
Theoremuhgrauhgrbi 39305 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
 |-  (
 ( G  e.  W  /\  V  =  (Vtx `  G )  /\  E  =  (iEdg `  G ) ) 
 ->  ( V UHGrph  E  <->  G  e. UHGraph  ) )
 
Theoremisuhgrop 39306 The property of being an undirected hypergraph 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, 1-Jan-2020.) (Revised by AV, 9-Oct-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( <. V ,  E >.  e. UHGraph 
 <->  E : dom  E --> ( ~P V  \  { (/)
 } ) ) )
 
Theoremuhgr0e 39307 The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
 |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  (iEdg `  G )  =  (/) )   =>    |-  ( ph  ->  G  e. UHGraph  )
 
Theoremuhgr0vb 39308 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.) (Revised by AV, 9-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  (/) )  ->  ( G  e. UHGraph  <->  (iEdg `  G )  =  (/) ) )
 
Theoremuhgr0 39309 The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
 |-  (/)  e. UHGraph
 
Theoremuhgrun 39310 The union  U of two (undirected) hypergraphs  G and  H with the same vertex set  V is a hypergraph with the vertex  V and the union  ( E  u.  F ) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.)
 |-  ( ph  ->  G  e. UHGraph  )   &    |-  ( ph  ->  H  e. UHGraph  )   &    |-  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. UHGraph  )
 
Theoremuhgrunop 39311 The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: 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, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)
 |-  ( ph  ->  G  e. UHGraph  )   &    |-  ( ph  ->  H  e. UHGraph  )   &    |-  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. UHGraph  )
 
Theoremushgrun 39312 The union  U of two (undirected) simple hypergraphs  G and  H with the same vertex set 
V is a (not necessarily simple) hypergraph with the vertex 
V and the union  ( E  u.  F
) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
 |-  ( ph  ->  G  e. USHGraph  )   &    |-  ( ph  ->  H  e. USHGraph  )   &    |-  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. UHGraph  )
 
Theoremushgrunop 39313 The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If  <. V ,  E >. and  <. V ,  F >. are simple hypergraphs, then  <. V ,  E  u.  F >. is a (not necessarily simple) hypergraph - 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. USHGraph  )   &    |-  ( ph  ->  H  e. USHGraph  )   &    |-  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. UHGraph  )
 
Theoremincistruhgr 39314* An incident structure 
<. P ,  L ,  I >. "where  P is a set whose elements are called points,  L is a distinct set whose elements are called lines and  I  C_  ( P  X.  L ) is the incidence relation" ( see Wikipedia "Incidence structure" (24-Oct-2020), https://en.wikipedia.org/wiki/Incidence_structure) implies an undirected hypergraph, if the incidence relation is right-total (to exclude empty edges). The points become the vertices, and the edge function is derived from the incidence relation by mapping each line ("edge") to the set of vertices incident to the line/edge. With  P  =  (
Base `  S ) and by defining two new slots for lines and incidence relations (analogous to LineG and Itv) and enhancing the definition of iEdg accordingly, it would even be possible to express that a corresponding incident structure is an undirected hypergraph. By choosing the incident relation appropriately, other kinds of undirected graphs (pseudographs, multigraphs, simple graphs, etc.) could be defined. (Contributed by AV, 24-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e.  W  /\  I  C_  ( P  X.  L )  /\  ran 
 I  =  L ) 
 ->  ( ( V  =  P  /\  E  =  ( e  e.  L  |->  { v  e.  P  |  v I e } )
 )  ->  G  e. UHGraph  ) )
 
21.33.8.4  Undirected pseudographs and multigraphs
 
Syntaxcupgr 39315 Extend class notation with undirected pseudographs.
 class UPGraph
 
Syntaxcumgr 39316 Extend class notation with undirected multigraphs.
 class UMGraph
 
Definitiondf-upgr 39317* Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set 
v (of "vertices") and a function  e (representing indexed "edges") 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. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgr 39318). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.)
 |- UPGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
 )  /  e ]. e : dom  e --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-umgr 39318* Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set 
v (of "vertices") and a function  e (representing indexed "edges") into subsets of  v of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." To provide uniform definitions for all kinds of graphs,  x  e.  ( ~P v  \  { (/) } ) is used as restriction of the class abstraction, although  x  e.  ~P v would be sufficient (see prprrab 39210 and isumgrs 39329). (Contributed by AV, 24-Nov-2020.)
 |- UMGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
 )  /  e ]. e : dom  e --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
 
Theoremisupgr 39319* The property of being an undirected pseudograph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e.  U  ->  ( G  e. UPGraph  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremwrdupgr 39320* The property of being an undirected pseudograph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e.  U  /\  E  e. Word  X )  ->  ( G  e. UPGraph  <->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremupgrf 39321* The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 39322 without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UPGraph  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremupgrfn 39322* The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremupgrss 39323 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 29-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F  e.  dom  E )  ->  ( E `  F ) 
 C_  V )
 
Theoremupgrn0 39324 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `
  F )  =/=  (/) )
 
Theoremupgrle 39325 An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  E  Fn  A  /\  F  e.  A )  ->  ( # `  ( E `  F ) )  <_  2 )
 
Theoremupgrfi 39326 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `
  F )  e. 
 Fin )
 
Theoremupgrex 39327* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  E  Fn  A  /\  F  e.  A )  ->  E. x  e.  V  E. y  e.  V  ( E `  F )  =  { x ,  y }
 )
 
Theoremisumgr 39328* The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e.  U  ->  ( G  e. UMGraph  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 ) )
 
Theoremisumgrs 39329* The simplified property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e.  U  ->  ( G  e. UMGraph  <->  E : dom  E --> { x  e.  ~P V  |  ( # `  x )  =  2 }
 ) )
 
Theoremwrdumgr 39330* The property of being an undirected multigraph, expressing the edges as "words". (Contributed by AV, 24-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e.  U  /\  E  e. Word  X )  ->  ( G  e. UMGraph  <->  E  e. Word  { x  e.  ~P V  |  ( # `  x )  =  2 } ) )
 
Theoremumgrf 39331* The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn 39332 without explicitly specified domain of the edge function. (Contributed by AV, 24-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UMGraph  ->  E : dom  E --> { x  e. 
 ~P V  |  ( # `  x )  =  2 } )
 
Theoremumgrfn 39332* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by AV, 24-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  E  Fn  A )  ->  E : A --> { x  e. 
 ~P V  |  ( # `  x )  =  2 } )
 
Theoremumgredg2 39333 An edge of a multigraph has exactly two ends. (Contributed by AV, 24-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  X  e.  dom  E )  ->  ( # `  ( E `
  X ) )  =  2 )
 
Theoremupgruhgr 39334 An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.)
 |-  ( G  e. UPGraph  ->  G  e. UHGraph  )
 
Theoremumgrupgr 39335 An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.)
 |-  ( G  e. UMGraph  ->  G  e. UPGraph  )
 
Theoremumgruhgr 39336 An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.)
 |-  ( G  e. UMGraph  ->  G  e. UHGraph  )
 
Theoremupgrle2 39337 An edge of an undirected pseudograph has at most two ends. (Contributed by AV, 6-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  X  e.  dom  I )  ->  ( # `  ( I `
  X ) ) 
 <_  2 )
 
Theoremumgrnloopv 39338 In a multigraph, 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, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  M  e.  W )  ->  (
 ( E `  X )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremumgredgprv 39339 In a multigraph, an edge is an unordered pair of vertices. This theorem would not hold for arbitrary hyper-/pseudographs since either  M or  N could be proper classes ( ( E `  X ) would be a loop in this case), which are no vertices of course. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. UMGraph  /\  X  e.  dom  E )  ->  ( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V ) ) )
 
Theoremumgrnloop 39340* In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UMGraph  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremumgrnloop0 39341* A multigraph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 11-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UMGraph  ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
 
Theoremumgr0e 39342 The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
 |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  (iEdg `  G )  =  (/) )   =>    |-  ( ph  ->  G  e. UMGraph  )
 
Theoremupgr0e 39343 The empty graph, with vertices but no edges, is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.)
 |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  (iEdg `  G )  =  (/) )   =>    |-  ( ph  ->  G  e. UPGraph  )
 
Theoremupgr1elem 39344* Lemma for upgr1e 39345 and uspgr1e 39465. (Contributed by AV, 16-Oct-2020.)
 |-  ( ph  ->  { B ,  C }  e.  S )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  { { B ,  C } }  C_  { x  e.  ( S  \  { (/)
 } )  |  ( # `  x )  <_ 
 2 } )
 
Theoremupgr1e 39345 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1e 39465. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 16-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { B ,  C } >. } )   =>    |-  ( ph  ->  G  e. UPGraph  )
 
Theoremupgr0eop 39346 The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, see usgr0eop 39467, and therefore also a multigraph ( G  e. UMGraph). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.)
 |-  ( V  e.  W  ->  <. V ,  (/) >.  e. UPGraph  )
 
Theoremupgr1eop 39347 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1eop 39468. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  <. V ,  { <. A ,  { B ,  C } >. } >.  e. UPGraph  )
 
Theoremupgr0eopALT 39348 Alternate proof of upgr0eop 39346, using the general theorem gropeld 39277 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 39346). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( V  e.  W  ->  <. V ,  (/) >.  e. UPGraph  )
 
Theoremupgr1eopALT 39349 Alternate proof of upgr1eop 39347, using the general theorem gropeld 39277 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr1eop 39347). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  <. V ,  { <. A ,  { B ,  C } >. } >.  e. UPGraph  )
 
Theoremupgrun 39350 The union  U of two 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, 12-Oct-2020.)
 |-  ( ph  ->  G  e. UPGraph  )   &    |-  ( ph  ->  H  e. UPGraph  )   &    |-  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  )
 
Theoremupgrunop 39351 The union of two pseudographs (with the same vertex set): If  <. V ,  E >. and  <. V ,  F >. are pseudographs, then  <. V ,  E  u.  F >. is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.)
 |-  ( ph  ->  G  e. UPGraph  )   &    |-  ( ph  ->  H  e. UPGraph  )   &    |-  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  )
 
Theoremumgrun 39352 The union  U of two multigraphs  G and  H with the same vertex set  V is a multigraph with the vertex  V and the union  ( E  u.  F ) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.)
 |-  ( ph  ->  G  e. UMGraph  )   &    |-  ( ph  ->  H  e. UMGraph  )   &    |-  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  )
 
Theoremumgrunop 39353 The union of two multigraphs (with the same vertex set): If  <. V ,  E >. and  <. V ,  F >. are multigraphs, then  <. V ,  E  u.  F >. is a multigraph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
 |-  ( ph  ->  G  e. UMGraph  )   &    |-  ( ph  ->  H  e. UMGraph  )   &    |-  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  )
 
21.33.8.5  Edges as subsets of vertices of graphs
 
Syntaxcedga 39354 Extend class notation with the set of edges (of an undirected simple (pseudo)graph) Remark: If this definition (and all related theorems) are moved to main.set, the label should become "cedg".
 class Edg
 
Definitiondf-edga 39355 Define the class of edges of a graph, see also definition "E = E(G)" in section I.1 of [Bollobas] p. 1. This definition is very general: It defines edges of a class as the range of its edge function (which even needs not to be a function). Therefore, this definition could also be used for hypergraphs, pseudographs and multigraphs. In these cases, however, the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. In some cases, this is no problem, so theorems with Edg are meaningful nevertheless (e.g., edguhgr 39365). Usually, however, this definition is used only for undirected simple (hyper-/pseudo-)graphs (with or without loops). (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
 |- Edg  =  ( g  e.  _V  |->  ran  (iEdg `  g )
 )
 
Theoremedgaval 39356 The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
 |-  ( G  e.  V  ->  (Edg `  G )  =  ran  (iEdg `  G ) )
 
Theoremedgaopval 39357 The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  (Edg `  <. V ,  E >. )  =  ran  E )
 
Theoremedgaov 39358 The edges of a graph represented as ordered pair, shown as operation value. Although a little less intuitive, this representation is often used because it is shorter than the representation as function value of a graph given as ordered pair, see edgopval 25123. The representation  ran  E for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 13-Oct-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( VEdg E )  =  ran  E )
 
Theoremedgastruct 39359 The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.)
 |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
 <. (.ef `  ndx ) ,  E >. }   =>    |-  ( ( V  e.  W  /\  E  e.  X )  ->  (Edg `  G )  =  ran  E )
 
Theoremedgiedgb 39360* A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e.  W  /\  Fun  I )  ->  ( E  e.  (Edg `  G )  <->  E. x  e.  dom  I  E  =  ( I `
  x ) ) )
 
Theoremuhgredgiedgb 39361* In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
 |-  I  =  (iEdg `  G )   =>    |-  ( G  e. UHGraph  ->  ( E  e.  (Edg `  G ) 
 <-> 
 E. x  e.  dom  I  E  =  ( I `
  x ) ) )
 
Theoremedg0iedg0 39362 There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.)
 |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e.  W  /\  Fun  I )  ->  ( E  =  (/)  <->  I  =  (/) ) )
 
Theoremuhgriedg0edg0 39363 A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 15-Dec-2020.)
 |-  ( G  e. UHGraph  ->  ( (Edg `  G )  =  (/)  <->  (iEdg `  G )  =  (/) ) )
 
Theoremuhgredgn0 39364 An edge of an undirected hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020.)
 |-  (
 ( G  e. UHGraph  /\  E  e.  (Edg `  G )
 )  ->  E  e.  ( ~P (Vtx `  G )  \  { (/) } )
 )
 
Theoremedguhgr 39365 An edge of an undirected hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.)
 |-  (
 ( G  e. UHGraph  /\  E  e.  (Edg `  G )
 )  ->  E  e.  ~P (Vtx `  G )
 )
 
Theoremuhgredgss 39366 The set of edges of a hypergraph is a subset of the powerset of vertices without the empty set. (Contributed by AV, 29-Nov-2020.)
 |-  ( G  e. UHGraph  ->  (Edg `  G )  C_  ( ~P (Vtx `  G )  \  { (/) } ) )
 
Theoremupgredgss 39367* The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020.)
 |-  ( G  e. UPGraph  ->  (Edg `  G )  C_  { x  e.  ( ~P (Vtx `  G )  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgredgss 39368* The set of edges of a multigraph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 25-Nov-2020.)
 |-  ( G  e. UMGraph  ->  (Edg `  G )  C_  { x  e.  ~P (Vtx `  G )  |  ( # `  x )  =  2 }
 )
 
Theoremedgupgr 39369 Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)
 |-  (
 ( G  e. UPGraph  /\  E  e.  (Edg `  G )
 )  ->  ( E  e.  ~P (Vtx `  G )  /\  E  =/=  (/)  /\  ( # `
  E )  <_ 
 2 ) )
 
Theoremedgumgr 39370 Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020.)
 |-  (
 ( G  e. UMGraph  /\  E  e.  (Edg `  G )
 )  ->  ( E  e.  ~P (Vtx `  G )  /\  ( # `  E )  =  2 )
 )
 
Theoremuhgrvtxedgiedgb 39371* In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  U  e.  V )  ->  ( E. i  e.  dom  I  U  e.  ( I `
  i )  <->  E. e  e.  E  U  e.  e )
 )
 
Theoremupgredg 39372* For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  C  e.  E )  ->  E. a  e.  V  E. b  e.  V  C  =  {
 a ,  b }
 )
 
Theoremumgredg 39373* For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  C  e.  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  C  =  { a ,  b } ) )
 
Theoremumgrpredgav 39374 An edge of a multigraph always connects two vertices. Analogue of umgredgprv 39339. This theorem does not hold for arbitrary pseudographs: if either  M or  N is a proper class, then  { M ,  N }  e.  E could still hold ( { M ,  N } would be either  { M } or  { N }, see prprc1 4095 or prprc2 4096, i.e. a loop), but  M  e.  V or  N  e.  V would not be true. (Contributed by AV, 27-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  { M ,  N }  e.  E )  ->  ( M  e.  V  /\  N  e.  V )
 )
 
Theoremupgredg2vtx 39375* For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 5-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  C  e.  E  /\  A  e.  C )  ->  E. b  e.  V  C  =  { A ,  b }
 )
 
Theoremupgredgpr 39376 If a proper pair (of vertices) is a subset of an edge in a pseudograph, the pair is the edge. (Contributed by AV, 30-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( ( G  e. UPGraph  /\  C  e.  E  /\  { A ,  B }  C_  C )  /\  ( A  e.  U  /\  B  e.  W  /\  A  =/=  B ) ) 
 ->  { A ,  B }  =  C )
 
Theoremumgredgne 39377 An edge of a multigraph always connects two different vertices. Analog of umgrnloopv 39338 resp. umgrnloop 39340. (Contributed by AV, 27-Nov-2020.)
 |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  { M ,  N }  e.  E )  ->  M  =/=  N )
 
Theoremumgrnloop2 39378 A multigraph has no loops. (Contributed by AV, 27-Oct-2020.) (Revised by AV, 30-Nov-2020.)
 |-  ( G  e. UMGraph  ->  { N ,  N }  e/  (Edg `  G ) )
 
21.33.8.6  Undirected simple graphs - basics

For undirected graphs, we will have the following hierarchy/taxonomy:

* Undirected Hypergraph: UHGraph

* Undirected simple Hypergraph: USHGraph => USHGraph  C_ UHGraph (ushgruhgr 39302)

* Undirected Pseudograph: UPGraph => UPGraph  C_ UHGraph (upgruhgr 39334)

* Undirected loop-free hypergraph: ULFHGraph (not defined formally yet) => ULFHGraph  C_ UHGraph

* Undirected loop-free simple hypergraph: ULFSHGraph (not defined formally yet) => ULFSHGraph  C_ USHGraph and ULFSHGraph  C_ ULFHGraph

* Undirected simple Pseudograph: USPGraph => USPGraph  C_ UPGraph (uspgrupgr 39407) and USPGraph  C_ USHGraph (uspgrushgr 39406), see also uspgrupgrushgr 39408

* Undirected Muligraph: UMGraph => UMGraph  C_ UPGraph (umgrupgr 39335) and UMGraph  C_ ULFHG (umgrislfupgr 39815)

* Undirected simple Graph: USGraph => USGraph  C_ USPGraph (usgruspgr 39409) and USGraph  C_ UMGraph (usgrumgr 39410) and USGraph  C_ ULFSHGraph (usgrislfuspgr 39816) see also usgrumgruspgr 39411

In this section, "simple graph" will always stand for "undirected simple graph (without loops)" and "simple pseudograph" for "undirected simple pseudograph (which could have loops)".

 
Syntaxcuspgr 39379 Extend class notation with undirected simple pseudographs (which could have loops).
 class USPGraph
 
Syntaxcusgr 39380 Extend class notation with undirected simple graphs (without loops).
 class USGraph
 
Definitiondf-uspgr 39381* Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph (see uspgrupgr 39407) or a special undirected simple hypergraph (see uspgrushgr 39406), consisting of a set  v (of "vertices") and an injective (one-to-one) function  e (representing (indexed) "edges") 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 pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
 |- USPGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
 )  /  e ]. e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-usgr 39382* Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr 39409), consisting of a set  v (of "vertices") and an injective (one-to-one) function  e (representing (indexed) "edges") into subsets of  v of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
 |- USGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
 )  /  e ]. e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
 
Theoremisuspgr 39383* The property of being a simple pseudograph. (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. USPGraph  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremisusgr 39384* 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 39385* 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 39386* 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 39387* The property of being a simple graph, simplified version of isusgr 39384. (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 39388* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 39386. (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 39389 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 39390 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 39391* 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 39392 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 39393* 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 39394 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 39395* 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 39396* 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 39397* 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 39398* 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 39399* 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 39400* 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  ) )
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