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Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuhgeq12gALTV 39301 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 39302 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.)
 |-  ( G  e. USHGraphALTV  ->  G  e. UHGraphALTV  )
 
Theoremuhguhgra 39303 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 39304 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 39305 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 39306 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 39307 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 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, 18-Jan-2020.)
 |-  (
 ( G  e.  W  /\  ( Base `  G )  =  (/) )  ->  ( G  e. UHGraphALTV  <->  (.ef `  G )  =  (/) ) )
 
Theoremuhgrepe 39309 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 39310 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 25049. Remark: The proof is much longer (although a lot is already covered by uhgrepe 39309) than the proof of the corresponding theorem uhgrares 25033 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 39311 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 25053. (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  )
 
21.33.9.5  Undirected hypergraphs (vertices)
 
Syntaxcvtxaltv 39312 Extend class notation with the vertices of undirected hypergraphs.
 class VtxALTV
 
Definitiondf-vtxALTV 39313 Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component. Therefore, this definition has no special meaning, except to give the first component of a graph its intended name. (Contributed by AV, 9-Jan-2020.)
 |- VtxALTV  =  ( g  e.  _V  |->  ( 1st `  g )
 )
 
Theoremvtxvalaltv 39314 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.)
 |-  ( G  e.  V  ->  ( VtxALTV  `  G )  =  ( 1st `  G )
 )
 
21.33.9.6  Size and order of undirected hypergraphs
 
Syntaxcgord 39315 Extend class notation with the order of undirected hypergraphs.
 class GrOrder
 
Syntaxcgsiz 39316 Extend class notation with the size of undirected hypergraphs.
 class GrSize
 
Definitiondf-gord 39317 Define the function mapping a graph to its order. This definition is very general: It defines the order for any ordered pair as the size of its first component. In the definition of [Bollobas] p. 3, there is no special symbol for the order of a graph G, it is simply denoted by |G| and defined by |G| = |V(G)|, which corresponds to our definition, see gordopval 39321: 
( GrOrder  `  <. V ,  E >. )  =  ( # `  V ). (Contributed by AV, 3-Jan-2020.)
 |- GrOrder  =  ( g  e.  _V  |->  ( # `  ( 1st `  g
 ) ) )
 
Definitiondf-gsiz 39318 Define the function mapping a graph to its size. This definition is very general: It defines the size for any ordered pair as the size of the domain of its second component (which even needs not to be a function). In the definition of [Bollobas] p. 3, the size of a graph G is denoted by e(G) (and implicitly defined by e(G) = |E(G)|, which corresponds to our definition, see uhgraopsiz 39323:  ( GrSize  `  <. V ,  E >. )  =  ( # `  dom  E ), or usgsizedg 39326:  ( GrSize  `  G
)  =  ( # `  ( Edges  `  G ) ) ). (Contributed by AV, 3-Jan-2020.)
 |- GrSize  =  ( g  e.  _V  |->  ( # `  dom  ( 2nd `  g ) ) )
 
Theoremgordval 39319 The order of a graph. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e.  V  ->  ( GrOrder  `  G )  =  ( # `  ( 1st `  G ) ) )
 
Theoremgsizval 39320 The size of a graph. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e.  V  ->  ( GrSize  `  G )  =  ( # `  dom  ( 2nd `  G ) ) )
 
Theoremgordopval 39321 The order of a graph represented as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( GrOrder  `  <. V ,  E >. )  =  ( # `  V ) )
 
Theoremgsizopval 39322 The size of a graph represented as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( GrSize  `  <. V ,  E >. )  =  ( # `  dom  E ) )
 
Theoremuhgraopsiz 39323 The size of an undirected hypergraph as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  ( V UHGrph  E  ->  ( GrSize  `  <. V ,  E >. )  =  ( # `  E ) )
 
Theoremuhgrasize 39324 The size of an undirected hypergraph represented as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  ( V UHGrph  E  ->  ( V GrSize  E )  =  ( # `  E ) )
 
Theoremuhgrasiz 39325 The size of an undirected hypergraph. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e. UHGrph  ->  ( GrSize  `  G )  =  ( # `  ( 2nd `  G ) ) )
 
21.33.9.7  Edges of undirected simple graphs without loops

Theorems of subsection "Undirected simple graphs - basics" expressed with the class Edges instead of the edge function. Conventions: Since only undirected simple graphs without loops are considered in this subsection, the term "graph" is used only in this restricted meaning. The labels for the theorems will contain the acronym "usg" instead of "usgra" for short. A graph by itself is usually denoted by the class variable  G ( G  e. USGrph). The class of edges is usually denoted by the class variable  E ( E  =  ( Edges  `  G )), while a class representing an edge by itself is usually denoted by the class variable  C ( C  e.  ( Edges  `  G )) , inspired by an edge being a *c*onnection between vertices. If a setvar variable is used for an edge, however, it is usually denoted by  e.

Acronyms used in labels: - ALT: alternative - usg: undirected simple graph without loops - edg: edge - pr: unordered pair - ppr: proper unordered pair, i e. an unordered pair containing to different sets - v: vertex - d: different - n: not - lp: loop - ad: adjacent - eu: existential uniqueness - imp: implication - inc: incident - f1: 1-1 function - lem: lemma - siz: size - ss: subset

 
Theoremusgsizedg 39326 The size of a graph is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.)
 |-  ( G  e. USGrph  ->  ( GrSize  `  G )  =  ( # `  ( Edges  `  G ) ) )
 
TheoremusgsizedgALT 39327 The size of a graph is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 12-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( G  e. USGrph  ->  ( # `  ( 2nd `  G ) )  =  ( # `
  ( Edges  `  G ) ) )
 
TheoremusgsizedgALT2 39328 The size of a graph is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  E  =  ( Edges  `  G )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( G  e. USGrph  ->  S  =  ( # `  E ) )
 
Theoremusgedgppr 39329 An edge of a graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge), analogous to umgrale 25046 and usgraedg2 25100. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  ( # `  C )  =  2 )
 
Theoremusgpredgv 39330 An edge of a graph always connects two vertices, analogous to usgraedgrnv 25102. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( VtxALTV  `  G )   =>    |-  ( ( G  e. USGrph  /\ 
 { M ,  N }  e.  E )  ->  ( M  e.  V  /\  N  e.  V ) )
 
TheoremusgpredgvALT 39331 An edge of a graph always connects two vertices, analogous to usgraedgrnv 25102. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 12-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\ 
 { M ,  N }  e.  E )  ->  ( M  e.  V  /\  N  e.  V ) )
 
Theoremedgssv2ALT 39332 An edge of a graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  ( C  C_  V  /\  ( # `  C )  =  2 )
 )
 
Theoremedgssv2 39333 An edge of a graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  ( C  C_  V  /\  ( # `  C )  =  2 )
 )
 
Theoremusgedgimp 39334* If there is an edge in a graph, there are two different vertices in the graph which are connected by this edge, analogous to usgraedg3 25111 and edgprvtx 25110 and usgrarnedg 25109. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( VtxALTV  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  E. x  e.  V  E. y  e.  V  ( x  =/=  y  /\  C  =  { x ,  y } ) )
 
Theoremusgvincvad 39335* If there is a vertex being incident with an edge in a graph, there is a(nother) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedg4 25112. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( VtxALTV  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E. y  e.  V  C  =  { X ,  y } )
 
Theoremusgvincvadeu 39336* If there is a vertex being incident with an edge in a graph, there is exactly one (other) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedgreu 25113. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( VtxALTV  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E! y  e.  V  C  =  { X ,  y } )
 
TheoremusgedgimpALT 39337* If there is an edge in a graph, there are two different vertices in the graph which are connected by this edge, analogous to usgraedg3 25111 and edgprvtx 25110 and usgrarnedg 25109. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  E. x  e.  V  E. y  e.  V  ( x  =/=  y  /\  C  =  { x ,  y } ) )
 
TheoremusgvincvadALT 39338* If there is a vertex being incident with an edge in a graph, there is a(nother) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedg4 25112. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E. y  e.  V  C  =  { X ,  y } )
 
TheoremusgvincvadeuALT 39339* If there is a vertex being incident with an edge in a graph, there is exactly one (other) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedgreu 25113. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E! y  e.  V  C  =  { X ,  y } )
 
Theoremusgpredgdv 39340 An edge of a graph always connects two different vertices, analogous to usgraedgrn 25106. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\ 
 { M ,  N }  e.  E )  ->  M  =/=  N )
 
Theoremusgedgnlp 39341* An edge of a graph is not a loop. (Contributed by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  -.  E. v  C  =  { v }
 )
 
Theoremusgvad2edg 39342* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex, analogous to usgra2edg 25107. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E )
 )  ->  E. x  e.  E  E. y  e.  E  ( x  =/=  y  /\  N  e.  x  /\  N  e.  y
 ) )
 
Theoremusg2edgneu 39343* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex, analogous to usgra2edg1 25108. Lemma for theorems about friendship graphs. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E )
 )  ->  -.  E! x  e.  E  N  e.  x )
 
Theoremusgedgvadf1lem1 39344* Lemma 1 for usgedgvadf1 39346. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V )
 
Theoremusgedgvadf1lem2 39345* Lemma 2 for usgedgvadf1 39346. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( M  =  (
 iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
 
Theoremusgedgvadf1 39346* The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 25118. The edge is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( e  e.  A  |->  (
 iota_ m  e.  V  e  =  { m ,  N } ) )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgedgvadf1ALTlem1 39347* Lemma 1 for usgedgvadf1 39346. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V )
 
Theoremusgedgvadf1ALTlem2 39348* Lemma 2 for usgedgvadf1 39346. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( M  =  (
 iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
 
Theoremusgedgvadf1ALT 39349* The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 25118. The edge is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( e  e.  A  |->  (
 iota_ m  e.  V  e  =  { m ,  N } ) )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgedgleord 39350* In a graph, the number of edges which contain a given vertex is not greater than the number of vertices, analogous to usgraedgleord 25119. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( # `  { e  e.  E  |  N  e.  e } )  <_  ( GrOrder  `  G ) )
 
TheoremusgedgleordALT 39351* Alternate version of usgedgleord 39350 with a shorter proof. In a graph, the number of edges which contain a given vertex is not greater than the order of the graph, i. e. the number of its vertices, analogous to usgraedgleord 25119. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 12-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  O  =  ( # `  V )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( # `  { e  e.  E  |  N  e.  e } )  <_  O )
 
21.33.9.8  Finite undirected simple graphs without loops
 
Syntaxcfusg 39352 Extend class notation with finite graphs.
 class FinUSGrph
 
Definitiondf-fusg 39353* Define the class of all finite undirected simple graphs without loops. Such a finite graph is an undirected simple graph without loops  <. V ,  E >. of finite order, i.e. where  V is finite. (Contributed by AV, 3-Jan-2020.)
 |- FinUSGrph  =  { <. v ,  e >.  |  ( e : dom  e -1-1-> { x  e.  ~P v  |  ( # `  x )  =  2 }  /\  v  e.  Fin ) }
 
Theoremrelfusgra 39354 The class of all finite undirected simple graph without loops is a relation. (Contributed by AV, 3-Jan-2020.)
 |-  Rel FinUSGrph
 
Theoremisfusgra 39355* The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V FinUSGrph  E  <->  ( E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }  /\  V  e.  Fin )
 ) )
 
Theoremisfusgra0 39356 The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V FinUSGrph  E  <->  ( V USGrph  E  /\  V  e.  Fin )
 ) )
 
Theoremisfusgracl 39357 The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e.  ( W  X.  X )  ->  ( G  e. FinUSGrph  <->  ( G  e. USGrph  /\  ( GrOrder  `  G )  e. 
 NN0 ) ) )
 
Theoremfusgraimpcl 39358 The implications of a finite undirected simple graph without loops. (Contributed by AV, 4-Jan-2020.)
 |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( GrOrder  `  G )  e.  NN0 ) )
 
TheoremisfusgraclALT 39359 The property of being a finite undirected simple graph without loops. (Contributed by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   =>    |-  ( G  e.  ( W  X.  X )  ->  ( G  e. FinUSGrph  <->  ( G  e. USGrph  /\  V  e.  Fin )
 ) )
 
TheoremfusgraimpclALT 39360 The implications of a finite undirected simple graph without loops. (Contributed by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   =>    |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  V  e.  Fin ) )
 
Theoremfusgusg 39361 A finite undirected simple graph without loops is a undirected simple graph without loops. (Contributed by AV, 16-Jan-2020.)
 |-  ( G  e. FinUSGrph  ->  G  e. USGrph  )
 
TheoremfusgraimpclALT2 39362 The implications of a finite undirected simple graph without loops. (Contributed by AV, 12-Jan-2020.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( 1st `  G )  e.  Fin ) )
 
Theoremfiusgedgfi 39363* In a finite graph the number of edges which contain a given vertex is also finite, analogous to fiusgraedgfi 25133. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  { e  e.  E  |  N  e.  e }  e.  Fin )
 
TheoremfiusgedgfiALT 39364* In a finite graph the number of edges which contain a given vertex is also finite, analogous to fiusgraedgfi 25133. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  { e  e.  E  |  N  e.  e }  e.  Fin )
 
21.33.9.9  Finite undirected simple graphs (extension)
 
Theoremusgedgffibi 39365 The number of edges in a graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.)
 |-  ( V USGrph  E  ->  ( E  e.  Fin  <->  ( V Edges  E )  e.  Fin ) )
 
Theoremusgo0s0 39366 The size of a graph of order 0 (i.e. with 0 vertices) is 0, analogous to usgrafisindb0 25134. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 4-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  0 )  ->  ( GrSize  `  G )  =  0 )
 
Theoremusgo0s0ALT 39367 The size of a graph of order 0 (i.e. with 0 vertices) is 0, analogous to usgrafisindb0 25134. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  0 ) 
 ->  S  =  0 )
 
Theoremusgo1s0ALT 39368 The size of a graph of order 1 (i.e. with 1 vertex) is 0, analogous to usgrafisindb1 25135. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  1 ) 
 ->  S  =  0 )
 
Theoremusgo0fis 39369 A graph of order 0 (i.e. with 0 vertices) has a finite set of edges, analogous to usgrafisbase 25140. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  0 )  ->  ( Edges  `  G )  e.  Fin )
 
Theoremusgo0fisALT 39370 A graph of order 0 (i.e. with 0 vertices) has a finite set of edges, analogous to usgrafisbase 25140. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  0 ) 
 ->  ( Edges  `  G )  e.  Fin )
 
TheoremusgrafisbaseALT 39371 Alternate version of usgrafisbase 25140, not depending on usgrafisindb0 25134. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  E  e.  Fin )
 
TheoremusgrafisbaseALT2 39372 Alternate version of usgrafisbase 25140, not depending on usgrafisindb0 25134. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Proof modification is discouraged.)
 |-  (
 ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  E  e.  Fin )
 
Theoremusgo1s0 39373 The size of a graph of order 1 (i.e. with 1 vertex) is 0, analogous to usgrafisindb1 25135. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 4-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  1 )  ->  ( GrSize  `  G )  =  0 )
 
Theoremusgresvm1 39374* Restricting an undirected simple graph by removing one vertex (and all edges ending at this vertex), analogous to usgrares1 25136. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( V  \  { N } ) USGrph  (  _I  |`  F ) )
 
Theoremusgfislem1 39375* Lemma 1 for usgfis 39377: The set of edges is the union of the edges containing a specific vertex and the edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 10-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  E  =  ( F  u.  { e  e.  E  |  N  e.  e } )
 
Theoremusgfislem2 39376* Lemma 2 for usgfis 39377: In a graph of finite order (i.e. with a finite number of vertices), the number of edges is finite if and only if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( VtxALTV  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  ( E  e.  Fin  <->  F  e.  Fin ) )
 
Theoremusgfis 39377 An undirected simple graph of finite order (i.e. with a finite number of vertices) is of finite size, i.e. it has also only a finite number of edges, analogous to usgrafis 25141. Remark: The proof of this theorem is very long compared with usgrafis 25141, because the theorem brfi1ind 12656 to perform the finite induction is taylored for binary relations, so that the theorem itself and the used lemmas must be transformed accordingly. Maybe a variant of brfi1ind 12656 could be provided, which is better suitable for this theorem. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 11-Jan-2018.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( GrSize  `  G )  e.  NN0 )
 
Theoremusgresvm1ALT 39378* Restricting an undirected simple graph by removing one vertex (and all edges ending at this vertex), analogous to usgrares1 25136. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( V  \  { N } ) USGrph  (  _I  |`  F ) )
 
TheoremusgfisALTlem1 39379* Lemma 1 for usgfisALT 39381: The set of edges is the union of the edges containing a specific vertex and the edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  E  =  ( F  u.  { e  e.  E  |  N  e.  e } )
 
TheoremusgfisALTlem2 39380* Lemma 2 for usgfis 39377: In a graph of finite order (i.e. with a finite number of vertices), the number of edges is finite if and only if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  ( E  e.  Fin  <->  F  e.  Fin ) )
 
TheoremusgfisALT 39381 An undirected simple graph of finite order (i.e. with a finite number of vertices) is of finite size, i.e. it has also only a finite number of edges, analogous to usgrafis 25141. Remark: The proof of this theorem is very long compared with usgrafis 25141, because the theorem brfi1ind 12656 to perform the finite induction is taylored for binary relations, so that the theorem itself and the used lemmas must be transformed accordingly. Maybe a variant of brfi1ind 12656 could be provided, which is better suitable for this theorem. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 15-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )
 
TheoremusgrafiedgALT 39382 A simple undirected graph with a finite number of vertices has also only a finite number of edges. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 4-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )
 
21.33.10  Monoids (extension)
 
21.33.10.1  Auxiliary theorems
 
Theoremovn0dmfun 39383 If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 5913. (Contributed by AV, 27-Jan-2020.)
 |-  (
 ( A F B )  =/=  (/)  ->  ( <. A ,  B >.  e.  dom  F 
 /\  Fun  ( F  |` 
 { <. A ,  B >. } ) ) )
 
Theoremxpsnopab 39384* A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)
 |-  ( { X }  X.  C )  =  { <. a ,  b >.  |  (
 a  =  X  /\  b  e.  C ) }
 
Theoremxpiun 39385* A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)
 |-  ( B  X.  C )  = 
 U_ x  e.  B  { <. a ,  b >.  |  ( a  =  x  /\  b  e.  C ) }
 
Theoremovn0ssdmfun 39386* If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 5913. (Contributed by AV, 27-Jan-2020.)
 |-  ( A. a  e.  D  A. b  e.  E  ( a F b )  =/=  (/)  ->  ( ( D  X.  E )  C_  dom 
 F  /\  Fun  ( F  |`  ( D  X.  E ) ) ) )
 
Theoremfnxpdmdm 39387 The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.)
 |-  ( F  Fn  ( A  X.  A )  ->  dom  dom  F  =  A )
 
Theoremcnfldsrngbas 39388 The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
 |-  R  =  (flds  S )   =>    |-  ( S  C_  CC  ->  S  =  ( Base `  R ) )
 
Theoremcnfldsrngadd 39389 The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
 |-  R  =  (flds  S )   =>    |-  ( S  e.  V  ->  +  =  ( +g  `  R ) )
 
Theoremcnfldsrngmul 39390 The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
 |-  R  =  (flds  S )   =>    |-  ( S  e.  V  ->  x.  =  ( .r
 `  R ) )
 
21.33.10.2  Magmas and Semigroups (extension)
 
Theoremplusfreseq 39391 If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .+^  =  ( +f `  M )   =>    |-  ( (/)  e/  ran  .+^  ->  (  .+  |`  ( B  X.  B ) )  =  .+^  )
 
Theoremmgmplusfreseq 39392 If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .+^  =  ( +f `  M )   =>    |-  ( ( M  e. Mgm  /\  (/)  e/  B )  ->  (  .+  |`  ( B  X.  B ) )  =  .+^  )
 
Theorem0mgm 39393 A set with an empty base set is always a magma". (Contributed by AV, 25-Feb-2020.)
 |-  ( Base `  M )  =  (/)   =>    |-  ( M  e.  V  ->  M  e. Mgm )
 
Theoremmgmpropd 39394* If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.)
 |-  ( ph  ->  B  =  (
 Base `  K ) )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  B  =/= 
 (/) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e. Mgm  <->  L  e. Mgm ) )
 
Theoremismgmd 39395* Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
 |-  ( ph  ->  B  =  (
 Base `  G ) )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  .+  =  ( +g  `  G ) )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  e.  B )   =>    |-  ( ph  ->  G  e. Mgm )
 
21.33.10.3  Magma homomorphisms and submagmas
 
Syntaxcmgmhm 39396 Hom-set generator class for magmas.
 class MgmHom
 
Syntaxcsubmgm 39397 Class function taking a magma to its lattice of submagmas.
 class SubMgm
 
Definitiondf-mgmhm 39398* A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.)
 |- MgmHom  =  ( s  e. Mgm ,  t  e. Mgm 
 |->  { f  e.  (
 ( Base `  t )  ^m  ( Base `  s )
 )  |  A. x  e.  ( Base `  s ) A. y  e.  ( Base `  s ) ( f `  ( x ( +g  `  s
 ) y ) )  =  ( ( f `
  x ) (
 +g  `  t )
 ( f `  y
 ) ) } )
 
Definitiondf-submgm 39399* A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.)
 |- SubMgm  =  ( s  e. Mgm  |->  { t  e.  ~P ( Base `  s
 )  |  A. x  e.  t  A. y  e.  t  ( x (
 +g  `  s )
 y )  e.  t } )
 
Theoremmgmhmrcl 39400 Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.)
 |-  ( F  e.  ( S MgmHom  T )  ->  ( S  e. Mgm  /\  T  e. Mgm )
 )
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40161
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