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Theorem List for Metamath Proof Explorer - 40101-40200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-conngr 40101* Define the class of all connected graphs. A graph is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv-av 40018 and dfconngr1 40102. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |- ConnGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. A. k  e.  v  A. n  e.  v  E. f E. p  f ( k (PathsOn `  g
 ) n ) p }
 
Theoremdfconngr1 40102* Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |- ConnGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. A. k  e.  v  A. n  e.  (
 v  \  { k } ) E. f E. p  f (
 k (PathsOn `  g ) n ) p }
 
Theoremisconngr 40103* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( G  e. ConnGraph  <->  A. k  e.  V  A. n  e.  V  E. f E. p  f ( k (PathsOn `  G ) n ) p ) )
 
Theoremisconngr1 40104* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( G  e. ConnGraph  <->  A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) E. f E. p  f ( k (PathsOn `  G ) n ) p ) )
 
Theoremcusconngr 40105 A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |-  (
 ( G  e. UHGraph  /\  G  e. ComplGraph )  ->  G  e. ConnGraph )
 
Theorem0conngr 40106 A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |-  (/)  e. ConnGraph
 
Theorem0vconngr 40107 A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  (/) )  ->  G  e. ConnGraph )
 
Theorem1conngr 40108 A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  { N } )  ->  G  e. ConnGraph )
 
21.33.8.22  Eulerian paths

According to Wikipedia ("Eulerian path", 9-Mar-2021, https://en.wikipedia.org/wiki/Eulerian_path): "In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. ... The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. ... A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian."

Correspondingly, an Eulerian path is defined as "a trail containing all edges" (see definition in [Bollobas] p. 16) in df-eupth 40110 resp. iseupth 40113.  (EulerPaths `  G
) is the set of all Eulerian paths in graph  G, see eupths 40112. An Eulerian circuit (called Euler tour in the definition in [Diestel] p. 22) is "a circuit in a graph containing all the edges" (see definition in [Bollobas] p. 16), or, with other words, a circuit which is an Eulerian path. The function mapping a graph to the set of its Eulerian paths is defined as EulerPaths in df-eupth 40110, whereas there is no explicit definition for Eulerian circuits (yet): The statement " <. F ,  P >. is an Eulerian circuit" is formally expressed by  ( F
(EulerPaths `  G ) P  /\  F (CircuitS `  G
) P ).

Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 40129). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 40157.

An Eulerian path does not have to be a path in the meaning of definition df-pths 39911, because it may traverse some vertices more than once. Therefore, "Eulerian trail" would be a more appropriate name.

The main result of this section is (one direction of) Euler's Theorem: "A non-trivial connected graph has an Euler[ian] circuit iff each vertex has even degree." (see part 1 of theorem 12 in [Bollobas] p. 16 and theorem 1.8.1 in [Diestel] p. 22) or, expressed with Eulerian paths: "A connected graph has an Euler[ian] trail from a vertex x to a vertex y (not equal with x) iff x and y are the only vertices of odd degree." (see part 2 of theorem 12 in [Bollobas] p. 17). In eulerpath 40153, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 40154 shows that a pseudograph with an Eulerian circuit has only vertices of even degree.

 
Syntaxceupth 40109 Extend class notation with Eulerian paths.
 class EulerPaths
 
Definitiondf-eupth 40110* Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |- EulerPaths  =  ( g  e.  _V  |->  {
 <. f ,  p >.  |  ( f (TrailS `  g
 ) p  /\  f : ( 0..^ ( # `  f ) )
 -onto->
 dom  (iEdg `  g )
 ) } )
 
Theoremreleupth 40111 The set  (EulerPaths `  G
) of all Eulerian paths on  G is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  Rel  (EulerPaths `
  G )
 
Theoremeupths 40112* The Eulerian paths on the graph  G. (Contributed by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  ( G  e.  X  ->  (EulerPaths `  G )  =  { <. f ,  p >.  |  ( f (TrailS `  G ) p  /\  f : ( 0..^ ( # `  f ) ) -onto-> dom 
 I ) } )
 
Theoremiseupth 40113 The property " <. F ,  P >. is an Eulerian path on the graph  G". An Eulerian path is defined as bijection  F from the edges to a set  0 ... ( N  -  1 ) and a function  P : ( 0 ... N ) --> V into the vertices such that for each  0  <_  k  <  N,  F ( k ) is an edge from  P ( k ) to  P ( k  +  1 ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e.  X  /\  F  e.  Y  /\  P  e.  Z )  ->  ( F (EulerPaths `  G ) P  <->  ( F (TrailS `  G ) P  /\  F : ( 0..^ ( # `  F ) )
 -onto->
 dom  I ) ) )
 
Theoremiseupthf1o 40114 The property " <. F ,  P >. is an Eulerian path on the graph  G". An Eulerian path is defined as bijection  F from the edges to a set  0 ... ( N  -  1 ) and a function  P : ( 0 ... N ) --> V into the vertices such that for each  0  <_  k  <  N,  F ( k ) is an edge from  P ( k ) to  P ( k  +  1 ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e.  X  /\  F  e.  Y  /\  P  e.  Z )  ->  ( F (EulerPaths `  G ) P  <->  ( F (1Walks `  G ) P  /\  F : ( 0..^ ( # `  F ) ) -1-1-onto-> dom 
 I ) ) )
 
Theoremeupthtrli 40115 Properties of an Eulerian path as a trail. (Contributed by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  ( F (EulerPaths `  G ) P 
 ->  ( F (TrailS `  G ) P  /\  F :
 ( 0..^ ( # `  F ) ) -onto-> dom 
 I ) )
 
Theoremeupthi 40116 Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  ( F (EulerPaths `  G ) P 
 ->  ( F (1Walks `  G ) P  /\  F :
 ( 0..^ ( # `  F ) ) -1-1-onto-> dom  I
 ) )
 
Theoremeupthf1o 40117 The  F function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  ( F (EulerPaths `  G ) P 
 ->  F : ( 0..^ ( # `  F ) ) -1-1-onto-> dom  I )
 
Theoremeupthfi 40118 Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  ( F (EulerPaths `  G ) P 
 ->  dom  I  e.  Fin )
 
Theoremeupthseg 40119 The  N-th edge in an eulerian path is the edge having  P ( N ) and  P ( N  +  1 ) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( F (EulerPaths `  G ) P  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  { ( P `  N ) ,  ( P `  ( N  +  1 ) ) }  C_  ( I `  ( F `  N ) ) )
 
Theoremupgriseupth 40120* The property " <. F ,  P >. is an Eulerian path on the pseudograph  G". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F  e.  U  /\  P  e.  Z )  ->  ( F (EulerPaths `  G ) P  <-> 
 ( F : ( 0..^ ( # `  F ) ) -1-1-onto-> dom  I  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( I `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theoremupgreupthi 40121* Properties of an Eulerian path in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F (EulerPaths `
  G ) P )  ->  ( F : ( 0..^ ( # `  F ) ) -1-1-onto-> dom 
 I  /\  P :
 ( 0 ... ( # `
  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( I `  ( F `
  k ) )  =  { ( P `
  k ) ,  ( P `  (
 k  +  1 ) ) } ) )
 
Theoremupgreupthseg 40122 The  N-th edge in an eulerian path is the edge from  P ( N ) to  P ( N  +  1 ). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F (EulerPaths `
  G ) P 
 /\  N  e.  (
 0..^ ( # `  F ) ) )  ->  ( I `  ( F `
  N ) )  =  { ( P `
  N ) ,  ( P `  ( N  +  1 )
 ) } )
 
Theoremeupthcl 40123 An Eulerian path has length 
# ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  ( F (EulerPaths `  G ) P 
 ->  ( # `  F )  e.  NN0 )
 
Theoremeupthistrl 40124 An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.)
 |-  ( F (EulerPaths `  G ) P 
 ->  F (TrailS `  G ) P )
 
Theoremeupthpf 40125 The  P function in an Eulerian path is a function from a finite sequence of nonnegative integers to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
 |-  ( F (EulerPaths `  G ) P 
 ->  P : ( 0
 ... ( # `  F ) ) --> (Vtx `  G ) )
 
Theoremeupth0 40126 There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  (
 ( A  e.  V  /\  I  =  (/) )  ->  (/) (EulerPaths `  G ) { <. 0 ,  A >. } )
 
Theoremeupthres 40127 The restriction  <. H ,  Q >. of an Eulerian path  <. F ,  P >. to an initial segment of the path (of length  N) forms an Eulerian path on the subgraph  S consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   &    |-  ( ph  ->  N  e.  ( 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  (iEdg `  S )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  H  =  ( F  |`  ( 0..^ N ) )   &    |-  Q  =  ( P  |`  ( 0
 ... N ) )   &    |-  (Vtx `  S )  =  V   =>    |-  ( ph  ->  H (EulerPaths `
  S ) Q )
 
Theoremeupthp1 40128 Append one path segment to an Eulerian path  <. F ,  P >. to become an Eulerian path  <. H ,  Q >. of the supergraph  S obtained by adding the new edge to the graph  G. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  -.  B  e.  dom 
 I )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  E  e.  (Edg `  G ) )   &    |-  ( ph  ->  { ( P `
  N ) ,  C }  C_  E )   &    |-  (iEdg `  S )  =  ( I  u.  { <. B ,  E >. } )   &    |-  H  =  ( F  u.  { <. N ,  B >. } )   &    |-  Q  =  ( P  u.  { <. ( N  +  1 ) ,  C >. } )   &    |-  (Vtx `  S )  =  V   &    |-  ( ( ph  /\  C  =  ( P `
  N ) ) 
 ->  E  =  { C } )   =>    |-  ( ph  ->  H (EulerPaths `
  S ) Q )
 
Theoremeupth2eucrct 40129 Append one path segment to an Eulerian path  <. F ,  P >. which may not be an (Eulerian) circuit to become an Eulerian circuit  <. H ,  Q >. of the supergraph  S obtained by adding the new edge to the graph  G. (Contributed by AV, 11-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  -.  B  e.  dom 
 I )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  E  e.  (Edg `  G ) )   &    |-  ( ph  ->  { ( P `
  N ) ,  C }  C_  E )   &    |-  (iEdg `  S )  =  ( I  u.  { <. B ,  E >. } )   &    |-  H  =  ( F  u.  { <. N ,  B >. } )   &    |-  Q  =  ( P  u.  { <. ( N  +  1 ) ,  C >. } )   &    |-  (Vtx `  S )  =  V   &    |-  ( ( ph  /\  C  =  ( P `
  N ) ) 
 ->  E  =  { C } )   &    |-  ( ph  ->  C  =  ( P `  0 ) )   =>    |-  ( ph  ->  ( H (EulerPaths `  S ) Q 
 /\  H (CircuitS `  S ) Q ) )
 
Theoremeupth2lem1 40130 TODO-AV: Duplicate of eupath2lem1 25784! Lemma for eupath2 25787. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/)
 ,  { A ,  B } )  <->  ( A  =/=  B 
 /\  ( U  =  A  \/  U  =  B ) ) ) )
 
Theoremeupth2lem2 40131 TODO-AV: Duplicate of eupath2lem2 25785! Lemma for eupath2 25787. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  B  e.  _V   =>    |-  ( ( B  =/=  C 
 /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
 ) 
 <->  U  e.  if ( A  =  C ,  (/)
 ,  { A ,  C } ) ) )
 
Theoremtrlsegvdeglem1 40132 Lemma for trlsegvdeg 40139. (Contributed by AV, 20-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   =>    |-  ( ph  ->  (
 ( P `  N )  e.  V  /\  ( P `  ( N  +  1 ) )  e.  V ) )
 
Theoremtrlsegvdeglem2 40133 Lemma for trlsegvdeg 40139. (Contributed by AV, 20-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  Fun  (iEdg `  X ) )
 
Theoremtrlsegvdeglem3 40134 Lemma for trlsegvdeg 40139. (Contributed by AV, 20-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  Fun  (iEdg `  Y ) )
 
Theoremtrlsegvdeglem4 40135 Lemma for trlsegvdeg 40139. (Contributed by AV, 21-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  dom  (iEdg `  X )  =  (
 ( F " (
 0..^ N ) )  i^i  dom  I )
 )
 
Theoremtrlsegvdeglem5 40136 Lemma for trlsegvdeg 40139. (Contributed by AV, 21-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  dom  (iEdg `  Y )  =  {
 ( F `  N ) } )
 
Theoremtrlsegvdeglem6 40137 Lemma for trlsegvdeg 40139. (Contributed by AV, 21-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  dom  (iEdg `  X )  e.  Fin )
 
Theoremtrlsegvdeglem7 40138 Lemma for trlsegvdeg 40139. (Contributed by AV, 21-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  dom  (iEdg `  Y )  e.  Fin )
 
Theoremtrlsegvdeg 40139 Formerly part of proof of eupath2lem3 25786: If a trail in a graph  G induces a subgraph  Z with the vertices  V of  G and the edges being the edges of the 1-walk, and a subgraph  X with the vertices  V of  G and the edges being the edges of the 1-walk except the last one, and a subgraph  Y with the vertices  V of  G and one edges being the last edge of the 1-walk, then the vertex degree of any vertex  U of  G within  Z is the sum of the vertex degree of  U within  X and the vertex degree of  U within  Y. Note that this theorem would not hold for arbitrary 1-walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  ( (VtxDeg `  Z ) `  U )  =  ( (
 (VtxDeg `  X ) `  U )  +  (
 (VtxDeg `  Y ) `  U ) ) )
 
Theoremeupth2lem3lem1 40140 Lemma for eupth2lem3 40148. (Contributed by AV, 21-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  ( (VtxDeg `  X ) `  U )  e.  NN0 )
 
Theoremeupth2lem3lem2 40141 Lemma for eupth2lem3 40148. (Contributed by AV, 21-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   =>    |-  ( ph  ->  ( (VtxDeg `  Y ) `  U )  e.  NN0 )
 
Theoremeupth2lem3lem3 40142* Lemma for eupth2lem3 40148, formerly part of proof of eupath2lem3 25786: If a loop  { ( P `
 N ) ,  ( P `  ( N  +  1 ) ) } is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  X ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   &    |-  ( ph  -> if- ( ( P `  N )  =  ( P `  ( N  +  1 ) ) ,  ( I `  ( F `  N ) )  =  { ( P `  N ) } ,  { ( P `  N ) ,  ( P `  ( N  +  1 ) ) }  C_  ( I `  ( F `  N ) ) ) )   =>    |-  ( ( ph  /\  ( P `  N )  =  ( P `  ( N  +  1 )
 ) )  ->  ( -.  2  ||  ( ( (VtxDeg `  X ) `  U )  +  (
 (VtxDeg `  Y ) `  U ) )  <->  U  e.  if ( ( P `  0 )  =  ( P `  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupth2lem3lem4 40143* Lemma for eupth2lem3 40148, formerly part of proof of eupath2lem3 25786: If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  X ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   &    |-  ( ph  -> if- ( ( P `  N )  =  ( P `  ( N  +  1 ) ) ,  ( I `  ( F `  N ) )  =  { ( P `  N ) } ,  { ( P `  N ) ,  ( P `  ( N  +  1 ) ) }  C_  ( I `  ( F `  N ) ) ) )   &    |-  ( ph  ->  ( I `  ( F `
  N ) )  e.  ~P V )   =>    |-  ( ( ph  /\  ( P `  N )  =/=  ( P `  ( N  +  1 )
 )  /\  ( U  =  ( P `  N )  \/  U  =  ( P `  ( N  +  1 ) ) ) )  ->  ( -.  2  ||  ( ( (VtxDeg `  X ) `  U )  +  (
 (VtxDeg `  Y ) `  U ) )  <->  U  e.  if ( ( P `  0 )  =  ( P `  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupth2lem3lem5 40144* Lemma for eupath2 25787. (Contributed by AV, 25-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  X ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   &    |-  ( ph  ->  ( I `  ( F `
  N ) )  =  { ( P `
  N ) ,  ( P `  ( N  +  1 )
 ) } )   =>    |-  ( ph  ->  ( I `  ( F `
  N ) )  e.  ~P V )
 
Theoremeupth2lem3lem6 40145* Formerly part of proof of eupath2lem3 25786: If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than  ( P `  0 ) and  ( P `  N ): if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  X ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   &    |-  ( ph  ->  ( I `  ( F `
  N ) )  =  { ( P `
  N ) ,  ( P `  ( N  +  1 )
 ) } )   =>    |-  ( ( ph  /\  ( P `  N )  =/=  ( P `  ( N  +  1
 ) )  /\  ( U  =/=  ( P `  N )  /\  U  =/=  ( P `  ( N  +  1 ) ) ) )  ->  ( -.  2  ||  ( ( (VtxDeg `  X ) `  U )  +  (
 (VtxDeg `  Y ) `  U ) )  <->  U  e.  if ( ( P `  0 )  =  ( P `  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupth2lem3lem7 40146* Lemma for eupath2lem3 25786: Combining trlsegvdeg 40139, eupth2lem3lem3 40142, eupth2lem3lem4 40143 and eupth2lem3lem6 40145. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  N  e.  (
 0..^ ( # `  F ) ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  F (TrailS `  G ) P )   &    |-  ( ph  ->  (Vtx `  X )  =  V )   &    |-  ( ph  ->  (Vtx `  Y )  =  V )   &    |-  ( ph  ->  (Vtx `  Z )  =  V )   &    |-  ( ph  ->  (iEdg `  X )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  ( ph  ->  (iEdg `  Y )  =  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } )   &    |-  ( ph  ->  (iEdg `  Z )  =  ( I  |`  ( F " ( 0 ... N ) ) ) )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  X ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   &    |-  ( ph  ->  ( I `  ( F `
  N ) )  =  { ( P `
  N ) ,  ( P `  ( N  +  1 )
 ) } )   =>    |-  ( ph  ->  ( -.  2  ||  (
 (VtxDeg `  Z ) `  U )  <->  U  e.  if ( ( P `  0 )  =  ( P `  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupthvdres 40147 Formerly part of proof of eupth2 40151: The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  Fun  I
 )   &    |-  ( ph  ->  F (EulerPaths `
  G ) P )   &    |-  H  =  <. V ,  ( I  |`  ( F " ( 0..^ ( # `  F ) ) ) )
 >.   =>    |-  ( ph  ->  (VtxDeg `  H )  =  (VtxDeg `  G ) )
 
Theoremeupth2lem3 40148* Lemma for eupath2 25787. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  G  e. UPGraph  )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   &    |-  H  =  <. V ,  ( I  |`  ( F " ( 0..^ N ) ) )
 >.   &    |-  X  =  <. V ,  ( I  |`  ( F
 " ( 0..^ ( N  +  1 ) ) ) ) >.   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( N  +  1 )  <_  ( # `  F ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  H ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   =>    |-  ( ph  ->  ( -.  2  ||  ( (VtxDeg `  X ) `  U ) 
 <->  U  e.  if (
 ( P `  0
 )  =  ( P `
  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupth2lemb 40149* Lemma for eupth2 40151 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). Formerly part of proof for eupth2 40151. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  G  e. UPGraph  )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   =>    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  <. V ,  ( I  |`  ( F
 " ( 0..^ 0 ) ) ) >. ) `
  x ) }  =  (/) )
 
Theoremeupth2lems 40150* Lemma for eupth2 40151 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. Formerly part of proof for eupth2 40151. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  G  e. UPGraph  )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   =>    |-  ( ( ph  /\  n  e.  NN0 )  ->  (
 ( n  <_  ( # `
  F )  ->  { x  e.  V  |  -.  2  ||  (
 (VtxDeg `  <. V ,  ( I  |`  ( F "
 ( 0..^ n ) ) ) >. ) `  x ) }  =  if ( ( P `  0 )  =  ( P `  n ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  n ) } ) )  ->  ( ( n  +  1 )  <_  ( # `  F )  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  <. V ,  ( I  |`  ( F
 " ( 0..^ ( n  +  1 ) ) ) ) >. ) `
  x ) }  =  if ( ( P `
  0 )  =  ( P `  ( n  +  1 )
 ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  ( n  +  1 ) ) }
 ) ) ) )
 
Theoremeupth2 40151* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  G  e. UPGraph  )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   =>    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  G ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  ( # `
  F ) ) ,  (/) ,  { ( P `  0 ) ,  ( P `  ( # `
  F ) ) } ) )
 
Theoremeulerpathpr 40152* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F (EulerPaths `
  G ) P )  ->  ( # `  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  G ) `  x ) }
 )  e.  { 0 ,  2 } )
 
Theoremeulerpath 40153* A pseudograph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. UPGraph  /\  (EulerPaths `  G )  =/=  (/) )  ->  ( # `  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  G ) `  x ) }
 )  e.  { 0 ,  2 } )
 
Theoremeulercrct 40154* A pseudograph with an Eulerian circuit  <. F ,  P >. (an "Eulerian pseudograph") has only vertices of even degree. (Contributed by AV, 12-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F (EulerPaths `
  G ) P 
 /\  F (CircuitS `  G ) P )  ->  A. x  e.  V  2  ||  (
 (VtxDeg `  G ) `  x ) )
 
Theoremeucrctshift 40155* Cyclically shifting the indices of an Eulerian circuit  <. F ,  P >. results in an Eulerian circuit  <. H ,  Q >.. (Contributed by AV, 15-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  Q  =  ( x  e.  ( 0
 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S )
 ) ,  ( P `
  ( ( x  +  S )  -  N ) ) ) )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   =>    |-  ( ph  ->  ( H (EulerPaths `  G ) Q 
 /\  H (CircuitS `  G ) Q ) )
 
Theoremeucrct2eupth1 40156 Removing one edge  ( I `  ( F `  N ) ) from a nonempty graph  G with an Eulerian circuit 
<. F ,  P >. results in a graph  S with an Eulerian path  <. H ,  Q >.. This is the special case of eucrct2eupth 40157 (with  J  =  ( N  -  1 )) where the last segment/edge of the circuit is removed. (Contributed by AV, 11-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  (Vtx `  S )  =  V   &    |-  ( ph  ->  0  <  ( # `  F ) )   &    |-  ( ph  ->  N  =  ( ( # `  F )  -  1
 ) )   &    |-  ( ph  ->  (iEdg `  S )  =  ( I  |`  ( F " ( 0..^ N ) ) ) )   &    |-  H  =  ( F  |`  ( 0..^ N ) )   &    |-  Q  =  ( P  |`  ( 0
 ... N ) )   =>    |-  ( ph  ->  H (EulerPaths `  S ) Q )
 
Theoremeucrct2eupth 40157* Removing one edge  ( I `  ( F `  J ) ) from a graph  G with an Eulerian circuit  <. F ,  P >. results in a graph  S with an Eulerian path  <. H ,  Q >.. (Contributed by AV, 17-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (EulerPaths `  G ) P )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  (Vtx `  S )  =  V   &    |-  ( ph  ->  N  =  ( # `  F ) )   &    |-  ( ph  ->  J  e.  ( 0..^ N ) )   &    |-  ( ph  ->  (iEdg `  S )  =  ( I  |`  ( F " ( ( 0..^ N )  \  { J }
 ) ) ) )   &    |-  K  =  ( J  +  1 )   &    |-  H  =  ( ( F cyclShift  K )  |`  ( 0..^ ( N  -  1 ) ) )   &    |-  Q  =  ( x  e.  ( 0..^ N )  |->  if ( x  <_  ( N  -  K ) ,  ( P `  ( x  +  K ) ) ,  ( P `  (
 ( x  +  K )  -  N ) ) ) )   =>    |-  ( ph  ->  H (EulerPaths `
  S ) Q )
 
21.33.8.23  The Königsberg Bridge problem

According to Wikipedia ("Seven Bridges of Königsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in [East] Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands - Kneiphof and Lomse - which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once.". Euler proved that the problem has no solution by applying Euler's theorem to the Königsberg graph, which is obtained by replacing each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which connects two land masses/vertices. The Königsberg graph  G is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair:  G  =  <. ( 0 ... 3
) ,  <" {
0 ,  1 } { 0 ,  2 } { 0 ,  3 } { 1 ,  2 } {
1 ,  2 }  { 2 ,  3 } {
2 ,  3 } "> >., see konigsbergumgr 40164. konigsberg-av 40171 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

 
Theoremkonigsbergvtx 40158 The set of vertices of the Königsberg graph  G. (Contributed by AV, 28-Feb-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  (Vtx `  G )  =  ( 0 ... 3
 )
 
Theoremkonigsbergiedg 40159 The indexed edges of the Königsberg graph  G. (Contributed by AV, 28-Feb-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  (iEdg `  G )  = 
 <" { 0 ,  1 }  { 0 ,  2 }  {
 0 ,  3 }  { 1 ,  2 }  { 1 ,  2 }  { 2 ,  3 }  {
 2 ,  3 } ">
 
Theoremkonigsbergiedgw 40160* The indexed edges of the Königsberg graph  G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  E  e. Word  { x  e. 
 ~P V  |  ( # `  x )  =  2 }
 
TheoremkonigsbergiedgwOLD 40161* The indexed edges of the Königsberg graph  G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) Obsolete version of konigsbergiedgw 40160 as of 9-Mar-2021. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }
 
Theoremkonigsbergssiedgwpr 40162* Each subset of the indexed edges of the Königsberg graph  G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  ( ( A  e. Word  _V 
 /\  B  e. Word  _V  /\  E  =  ( A ++ 
 B ) )  ->  A  e. Word  { x  e. 
 ~P V  |  ( # `  x )  =  2 } )
 
Theoremkonigsbergssiedgw 40163* Each subset of the indexed edges of the Königsberg graph  G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  ( ( A  e. Word  _V 
 /\  B  e. Word  _V  /\  E  =  ( A ++ 
 B ) )  ->  A  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremkonigsbergumgr 40164 The Königsberg graph  G is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  G  e. UMGraph
 
TheoremkonigsbergupgrOLD 40165 The Königsberg graph  G is a pseudograph. (Contributed by AV, 28-Feb-2021.) Obsolete version of konigsbergumgr 40164 as of 9-Mar-2021. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  G  e. UPGraph
 
Theoremkonigsberglem1 40166 Lemma 1 for konigsberg-av 40171: Vertex  0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  ( (VtxDeg `  G ) `  0 )  =  3
 
Theoremkonigsberglem2 40167 Lemma 2 for konigsberg-av 40171: Vertex  1 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  ( (VtxDeg `  G ) `  1 )  =  3
 
Theoremkonigsberglem3 40168 Lemma 3 for konigsberg-av 40171: Vertex  3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  ( (VtxDeg `  G ) `  3 )  =  3
 
Theoremkonigsberglem4 40169* Lemma 4 for konigsberg-av 40171: Vertices  0 ,  1 ,  3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  { 0 ,  1 ,  3 }  C_  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  G ) `  x ) }
 
Theoremkonigsberglem5 40170* Lemma 5 for konigsberg-av 40171: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  2  <  ( # `  { x  e.  V  |  -.  2  ||  ( (VtxDeg `  G ) `  x ) }
 )
 
Theoremkonigsberg-av 40171 The Königsberg Bridge problem. If  G is the Königsberg graph, i.e. a graph on four vertices  0 ,  1 , 
2 ,  3, with edges  { 0 ,  1 } ,  {
0 ,  2 } ,  { 0 ,  3 } ,  {
1 ,  2 } ,  { 1 ,  2 } ,  { 2 ,  3 } ,  {
2 ,  3 }, then vertices  0 ,  1 , 
3 each have degree three, and  2 has degree five, so there are four vertices of odd degree and thus by eupath 25788 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  (EulerPaths `
  G )  =  (/)
 
21.33.9  Graph theory (old)
 
21.33.9.1  Undirected hypergraphs
 
Theoremuhgraedgrnv 40172 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
 
Theoremusgra2pthspth 40173 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 40174 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 40175* Lemma for usgra2pth 40176. (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 40176* 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 40177* 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 40178 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 40179* 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 40180 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 40181* 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 40182 Extend class notation with undirected hypergraphs as extensible structures.
 class UHGraphALTV
 
Syntaxcushgraltv 40183 Extend class notation with undirected simple hypergraphs as extensible structures.
 class USHGraphALTV
 
Definitiondf-uhgrALTV 40184* 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 40185* 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 40186 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 40187 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 40188 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 40189 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 )
 
Theoremuhgeq12gALTV 40190 If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  (.ef `  G )   &    |-  W  =  ( Base `  H )   &    |-  F  =  (.ef `  H )   =>    |-  (
 ( ( G  e.  X  /\  H  e.  Y )  /\  ( V  =  W  /\  E  =  F ) )  ->  ( G  e. UHGraphALTV 
 <->  H  e. UHGraphALTV  ) )
 
Theoremushguhg 40191 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.)
 |-  ( G  e. USHGraphALTV  ->  G  e. UHGraphALTV  )
 
Theoremuhguhgra 40192 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.)
 |-  (
 ( G  e. UHGraphALTV  /\  V  =  ( Base `  G )  /\  E  =  (.ef `  G ) )  ->  V UHGrph  E )
 
Theoremuhgrauhg 40193 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.)
 |-  (
 ( V UHGrph  E  /\  V  =  ( Base `  G )  /\  E  =  (.ef `  G )
 )  ->  ( G  e.  W  ->  G  e. UHGraphALTV  ) )
 
Theoremuhgrauhgbi 40194 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.)
 |-  (
 ( G  e.  W  /\  V  =  ( Base `  G )  /\  E  =  (.ef `  G )
 )  ->  ( V UHGrph  E  <->  G  e. UHGraphALTV  ) )
 
Theoremuhgeq12d 40195 If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph, deduction form. (Contributed by AV, 18-Jan-2020.)
 |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  ( ph  ->  (.ef `  G )  =  (.ef `  H ) )   &    |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  H  e.  Y )   =>    |-  ( ph  ->  ( G  e. UHGraphALTV  <->  H  e. UHGraphALTV  ) )
 
Theoremuhg0e 40196 The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  (
 ( G  e.  W  /\  (.ef `  G )  =  (/) )  ->  G  e. UHGraphALTV  )
 
Theoremuhg0v 40197 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, 18-Jan-2020.)
 |-  (
 ( G  e.  W  /\  ( Base `  G )  =  (/) )  ->  ( G  e. UHGraphALTV  <->  (.ef `  G )  =  (/) ) )
 
Theoremuhgrepe 40198 Replacing the edges of a hypergraph results in a hypergraph. (Contributed by AV, 18-Jan-2020.) (Proof shortened by AV, 24-Oct-2020.)
 |-  V  =  ( Base `  G )   &    |-  S  =  (.ef `  ndx )   &    |-  ( ph  ->  G  e. UHGraphALTV  )   &    |-  ( ph  ->  E : dom  E --> ( ~P V  \  { (/) } ) )   &    |-  ( ph  ->  E  e.  _V )   =>    |-  ( ph  ->  ( G sSet  <. S ,  E >. )  e. UHGraphALTV  )
 
Theoremuhgres 40199 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 25130. Remark: The proof is much longer (although a lot is already covered by uhgrepe 40198) than the proof of the corresponding theorem uhgrares 25114 for graphs defined as pairs. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  E  =  (.ef `  G )   =>    |-  (
 ( G  e. UHGraphALTV  /\  F  =  ( E  |`  A ) )  ->  ( G sSet  <.
 (.ef `  ndx ) ,  F >. )  e. UHGraphALTV  )
 
Theoremuhgun 40200 The union of two (undirected) hypergraphs (with the same vertex set): 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), analogous to umgraun 25134. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  ( ph  ->  G  e. UHGraphALTV  )   &    |-  ( ph  ->  H  e. UHGraphALTV  )   &    |-  E  =  (.ef `  G )   &    |-  F  =  (.ef `  H )   &    |-  S  =  (.ef `  ndx )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( G sSet  <. S ,  ( E  u.  F ) >. )  e. UHGraphALTV  )
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