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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpt2xopoveq 28201* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )   =>    |-  ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [.
 <. V ,  W >.  /  x ]. [. K  /  y ]. ph } )
 
Theoremmpt2xopovel 28202* Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )   =>    |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  ( <. V ,  W >. F K )  <->  ( K  e.  V  /\  N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
 ) )
 
Theoremmpt2xopoveqd 28203* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )   &    |-  ( ps  ->  ( V  e.  X  /\  W  e.  Y )
 )   &    |-  ( ( ps  /\  -.  K  e.  V ) 
 ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  =  (/) )   =>    |-  ( ps  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [.
 <. V ,  W >.  /  x ]. [. K  /  y ]. ph } )
 
Theoremmpt2ndm0 28204* The value of an operation given by a maps-to rule is the empty set if the arguments ar not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )   =>    |-  ( -.  ( V  e.  X  /\  W  e.  Y )  ->  ( V F W )  =  (/) )
 
Theorembrovex 28205* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  C )   &    |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  Rel  ( V O E ) )   =>    |-  ( F ( V O E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
 ) )
 
Theorembrovmpt2ex 28206* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  {
 <. z ,  w >.  | 
 ph } )   =>    |-  ( F ( V O E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
 
Theoremsprmpt2 28207* The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  M  =  ( v  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( f ( v W e ) p 
 /\  ch ) } )   &    |-  (
 ( v  =  V  /\  e  =  E )  ->  ( ch  <->  ps ) )   &    |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V W E ) p 
 ->  th ) )   &    |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )   =>    |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  ( V M E )  =  { <. f ,  p >.  |  ( f ( V W E ) p 
 /\  ps ) } )
 
Theoremisprmpt2 28208* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p 
 /\  ps ) } )   &    |-  (
 ( f  =  F  /\  p  =  P )  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ( F  e.  X  /\  P  e.  Y ) 
 ->  ( F M P  <->  ( F W P  /\  ch ) ) ) )
 
18.23.3.7  Half-open integer ranges (extension)
 
Theoremfzossrbm1 28209 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  ( N  e.  NN0  ->  (
 0..^ ( N  -  1 ) )  C_  ( 0..^ N ) )
 
Theoremfzo0to3tp 28210 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 0..^ 3 )  =  { 0 ,  1 ,  2 }
 
Theoremfzo0to42pr 28211 A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
 |-  (
 0..^ 4 )  =  ( { 0 ,  1 }  u.  {
 2 ,  3 } )
 
Theoremfzon 28212 A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <_  M  <->  ( M..^ N )  =  (/) ) )
 
Theoremelfznelfzo 28213 A value in a finite set of sequential integers is a border value if it is not contained in the half-open integer range contained in the finite set of sequential integerss. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  (
 ( y  e.  (
 0 ... K )  /\  -.  y  e.  ( 1..^ K ) )  ->  ( y  =  0  \/  y  =  K ) )
 
Theoreminjresinjlem 28214 Lemma for injresinj 28215. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( -.  y  e.  (
 1..^ K )  ->  ( ( F `  0 )  =/=  ( F `  K )  ->  ( ( F :
 ( 0 ... K )
 --> V  /\  K  e.  NN0 )  ->  ( (
 ( F " {
 0 ,  K }
 )  i^i  ( F " ( 1..^ K ) ) )  =  (/)  ->  ( ( x  e.  ( 0 ... K )  /\  y  e.  (
 0 ... K ) ) 
 ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
 ) ) ) ) )
 
Theoreminjresinj 28215 A function whose restriction is injective and the values of the remaining arguments are different from all other values is injective itself. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( K  e.  NN0  ->  (
 ( F : ( 0 ... K ) --> V  /\  Fun  `' ( F  |`  ( 1..^ K ) )  /\  ( F `  0 )  =/=  ( F `  K ) )  ->  ( ( ( F
 " { 0 ,  K } )  i^i  ( F " (
 1..^ K ) ) )  =  (/)  ->  Fun  `' F ) ) )
 
18.23.3.8  The ` # ` (finite set size) function (extension)
 
Theoremelprchashprn2 28216 If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  ( -.  M  e.  _V  ->  -.  ( # `  { M ,  N } )  =  2 )
 
Theoremhashtpg 28217 The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( ( A  =/=  B 
 /\  B  =/=  C  /\  C  =/=  A )  <-> 
 ( # `  { A ,  B ,  C }
 )  =  3 ) )
 
Theoremhashgt12el 28218* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
 |-  (
 ( V  e.  W  /\  1  <  ( # `  V ) )  ->  E. a  e.  V  E. b  e.  V  a  =/=  b )
 
Theoremhashgt12el2 28219* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
 |-  (
 ( V  e.  W  /\  1  <  ( # `  V )  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b
 )
 
18.23.3.9  Words over a set (extension)
 
Theorem4fvwrd4 28220* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  (
 ( L  e.  ( ZZ>=
 `  3 )  /\  P : ( 0 ...
 L ) --> V ) 
 ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( P `
  0 )  =  a  /\  ( P `
  1 )  =  b )  /\  (
 ( P `  2
 )  =  c  /\  ( P `  3 )  =  d ) ) )
 
18.23.3.10  Longer string literals (extension)
 
Theorems2prop 28221 A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( A  e.  S  /\  B  e.  S ) 
 ->  <" A B ">  =  { <. 0 ,  A >. ,  <. 1 ,  B >. } )
 
Theorems4prop 28222 A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B C D ">  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
 <. 3 ,  D >. } ) )
 
Theorems2f1o 28223 A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  ->  ( E  =  <" A B ">  ->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )
 
Theorems4dom 28224 The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  (
 ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  ( E  =  <" A B C D ">  ->  dom 
 E  =  ( {
 0 ,  1 }  u.  { 2 ,  3 } ) ) )
 
Theorems4f1o 28225 A length 4 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D ) )  ->  ( E  =  <" A B C D ">  ->  E : dom  E -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) ) ) )
 
18.23.3.11  Undirected simple graphs

Although undirected simple graphs (with or without loops) are defined separately from undirected mulitigraphs (see df-umgra 23878), the definitions are similar and therefore compatible with each other, see uslisumgra 28246 and usisuslgra 28247.

 
Syntaxcuslg 28226 Extend class notation with undirected (simple) graphs with loops.
 class USLGrph
 
Syntaxcusg 28227 Extend class notation with undirected (simple) graphs (without loops).
 class USGrph
 
Definitiondf-uslgra 28228* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-usgra 28229* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph  <. V ,  E >. where 
E is an injective (one-to-one) function into subsets of  V of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itsself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
 
Theoremreluslgra 28230 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  Rel USLGrph
 
Theoremrelusgra 28231 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  Rel USGrph
 
Theoremuslgrav 28232 The classes of vertices and edges of an undirected simple graph with loops are sets. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USLGrph  E  ->  ( V  e.  _V  /\  E  e.  _V ) )
 
Theoremusgrav 28233 The classes of vertices and edges of an undirected simple graph without loops are sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( V USGrph  E  ->  ( V  e.  _V  /\  E  e.  _V ) )
 
Theoremisuslgra 28234* The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V USLGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremisusgra 28235* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 ) )
 
Theoremuslgraf 28236* The edge function of an undirected simple graph with loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremusgraf 28237* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 )
 
Theoremisusgra0 28238* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 ) )
 
Theoremusgraf0 28239* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 )
 
Theoremusgrafun 28240 The edge function of an undirected simple graph without loops is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.)
 |-  ( V USGrph  E  ->  Fun  E )
 
Theoremusgraedgop 28241 An edge of an undirected simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  X  e.  dom  E ) 
 ->  ( ( E `  X )  =  { M ,  N }  <->  <. X ,  { M ,  N } >.  e.  E ) )
 
Theoremusgraf1o 28242 The edge function of an undirected simple graph without loops is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran  E )
 
Theoremusgraf1 28243 The edge function of an undirected simple graph without loops is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> ran  E )
 
Theoremusgrass 28244 An edge is a subset of vertices, analogous to umgrass 23886. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  F  e.  dom  E ) 
 ->  ( E `  F )  C_  V )
 
Theoremusgraeq12d 28245 Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F ) )  ->  ( V USGrph  E 
 <->  W USGrph  F ) )
 
Theoremuslisumgra 28246 An undirected simple graph with loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  V UMGrph  E )
 
Theoremusisuslgra 28247 An undirected simple graph without loops is an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  V USLGrph  E )
 
Theoremusisumgra 28248 An undirected simple graph without loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USGrph  E  ->  V UMGrph  E )
 
Theoremusgrares 28249 A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 23891. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  V USGrph  ( E  |`  A ) )
 
Theoremusgra0 28250 The empty graph, with vertices but no edges, is a graph, analogous to umgra0 23892. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V  e.  W  ->  V USGrph  (/) )
 
Theoremusgra0v 28251 The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  ( (/) USGrph  E 
 <->  E  =  (/) )
 
Theoremuslgra1 28252 The graph with one edge, analogous to umgra1 23893. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  V USLGrph  { <. A ,  { B ,  C } >. } )
 
Theoremusgra1 28253 The graph with one edge, analogous to umgra1 23893 ( with additional assumption that  B  =/=  C since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  ( B  =/=  C  ->  V USGrph  {
 <. A ,  { B ,  C } >. } )
 )
 
Theoremuslgraun 28254 If  <. V ,  E >. and  <. V ,  F >. are (simple) graphs (with loops), then  <. V ,  E  u.  F >. is a multigraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices), analogous to umgraun 23894. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V USLGrph  E )   &    |-  ( ph  ->  V USLGrph  F )   =>    |-  ( ph  ->  V UMGrph  ( E  u.  F ) )
 
Theoremusgraedg2 28255 The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 23888. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  X  e.  dom  E ) 
 ->  ( # `  ( E `  X ) )  =  2 )
 
Theoremusgraedgprv 28256 In an undirected graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  X  e.  dom  E ) 
 ->  ( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V ) ) )
 
Theoremusgraedgrnv 28257 An edge of an undirected simple graph always connects to vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  { M ,  N }  e.  ran  E )  ->  ( M  e.  V  /\  N  e.  V ) )
 
Theoremusgranloop 28258* In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgraedgrn 28259 An edge of an undirected simple graph without loops always connects two different vertices. (Contributed by Alexander van der Vekens, 2-Sep-2017.)
 |-  (
 ( V USGrph  E  /\  { M ,  N }  e.  ran  E )  ->  M  =/=  N )
 
18.23.3.12  Undirected simple graphs (examples)
 
Theoremusgra1v 28260 A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( { A } USGrph  E  <->  E  =  (/) )
 
Theoremusgraexvlem 28261 Lemma for usgraexmpl 28267. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  V  =  ( {
 0 ,  1 ,  2 }  u.  {
 3 ,  4 } )
 
Theoremusgraex0elv 28262 Lemma 0 for usgraexmpl 28267. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  0  e.  V
 
Theoremusgraex1elv 28263 Lemma 1 for usgraexmpl 28267. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  1  e.  V
 
Theoremusgraex2elv 28264 Lemma 2 for usgraexmpl 28267. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  2  e.  V
 
Theoremusgraex3elv 28265 Lemma 3 for usgraexmpl 28267. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  3  e.  V
 
Theoremusgraexmpldifpr 28266 Lemma for usgraexmpl 28267: all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  (
 ( { 0 ,  1 }  =/=  {
 1 ,  2 } 
 /\  { 0 ,  1 }  =/=  { 2 ,  0 }  /\  { 0 ,  1 }  =/=  { 0 ,  3 } )  /\  ( { 1 ,  2 }  =/=  { 2 ,  0 }  /\  { 1 ,  2 }  =/=  { 0 ,  3 }  /\  {
 2 ,  0 }  =/=  { 0 ,  3 } ) )
 
Theoremusgraexmpl 28267  <. V ,  E >. is a graph of five vertices  0 ,  1 , 
2 ,  3 ,  4, with edges  { 0 ,  1 } ,  {
1 ,  2 } ,  { 2 ,  0 } ,  {
0 ,  3 }. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   =>    |-  V USGrph  E
 
18.23.3.13  Neighbors, complete graphs and universal vertices
 
Syntaxcnbgra 28268 Extend class notation with Neighbors (of a vertex in a graph).
 class Neighbors
 
Syntaxccusgra 28269 Extend class notation with complete (undirected simple) graphs.
 class ComplUSGrph
 
Syntaxcuvtx 28270 Extend class notation with the universal vertices (in a graph).
 class UnivVertex
 
Definitiondf-nbgra 28271* Define the class of all Neighbors of a vertex in a graph. The neighbors of a vertex are all vertices which are connected with this vertex by an edge. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.)
 |- Neighbors  =  ( g  e.  _V ,  k  e.  ( 1st `  g )  |->  { n  e.  ( 1st `  g
 )  |  { k ,  n }  e.  ran  ( 2nd `  g ) } )
 
Definitiondf-cusgra 28272* Define the class of all complete (undirected simple) graphs. An undirected simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v  A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
 
Definitiondf-uvtx 28273* Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |- UnivVertex  =  ( v  e.  _V ,  e  e.  _V  |->  { n  e.  v  |  A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
 
Theoremnbgraop 28274* The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  (
 ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
 
Theoremnbgrael 28275 The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( N  e.  ( <. V ,  E >. Neighbors  K ) 
 <->  ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E ) ) )
 
Theoremnbgranv0 28276 There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( N  e/  V  ->  ( <. V ,  E >. Neighbors  N )  =  (/) )
 
Theoremnbusgra 28277* The set of neighbors of a vertex in a graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.)
 |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
 
Theoremnbgra0nb 28278* A vertex which is not endpoint of an edge has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  ( <. V ,  E >. Neighbors  N )  =  (/) ) )
 
Theoremnbgraeledg 28279 A class/vertex is a neighbor of another class/vertex if and only if it is an endpoint of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  { N ,  K }  e.  ran  E ) )
 
Theoremnbgraisvtx 28280 Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  ->  N  e.  V ) )
 
Theoremnbgra0edg 28281 In a graph with no edges, every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph 
 (/)  ->  ( <. V ,  (/)
 >. Neighbors  K )  =  (/) )
 
Theoremnbgrassvt 28282 The neighbors of a node in a graph are a subset of all nodes of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N ) 
 C_  V )
 
Theoremnbgranself 28283* A node in a graph (without loops!) is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  A. v  e.  V  v  e/  ( <. V ,  E >. Neighbors  v
 ) )
 
Theoremnbgrassovt 28284 The neighbors of a vertex are a subset of the other vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  V  ->  ( <. V ,  E >. Neighbors  N ) 
 C_  ( V  \  { N } ) ) )
 
Theoremnbgranself2 28285 A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  N  e/  ( <. V ,  E >. Neighbors  N ) )
 
Theoremnbgrasym 28286 A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  K  e.  ( <. V ,  E >. Neighbors  N ) ) )
 
Theoremnb3graprlem1 28287 Lemma 1 for nb3grapr 28289. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( (
 <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E 
 /\  { A ,  C }  e.  ran  E ) ) )
 
Theoremnb3graprlem2 28288* Lemma 2 for nb3grapr 28289. (Contributed by Alexander van der Vekens, 17-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v }
 ) ( <. V ,  E >. Neighbors  A )  =  {
 v ,  w }
 ) )
 
Theoremnb3grapr 28289* The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y }
 ) ( <. V ,  E >. Neighbors  x )  =  {
 y ,  z }
 ) )
 
Theoremnb3grapr2 28290 The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B } ) ) )
 
Theoremnb3gra2nb 28291 If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }
 ) 
 <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
 ) ) )
 
Theoremiscusgra 28292* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V ComplUSGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) { n ,  k }  e.  ran  E ) ) )
 
Theoremiscusgra0 28293* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  ( V USGrph  E 
 /\  A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) { n ,  k }  e.  ran  E ) )
 
Theoremcusisusgra 28294 A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  V USGrph  E )
 
Theoremcusgra0v 28295 A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  (/) ComplUSGrph  (/)
 
Theoremcusgra1v 28296 A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  { A } ComplUSGrph  (/)
 
Theoremcusgra2v 28297 A graph with two (different) vertices is complete if and only if there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  ( { A ,  B } USGrph  E  ->  ( { A ,  B } ComplUSGrph  E  <->  { A ,  B }  e.  ran  E ) ) )
 
Theoremnbcusgra 28298 In a complete (undirected simple) graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )
 )
 
Theoremcusgra3v 28299 A graph with three (different) vertices is complete if and only if there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  V  =  { A ,  B ,  C }   =>    |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  V USGrph  E  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 ->  ( V ComplUSGrph  E  <->  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )
 
Theoremcusgra3vnbpr 28300* The neighbors of a vertex in a graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
 |-  V  =  { A ,  B ,  C }   =>    |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  V USGrph  E  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 ->  ( V ComplUSGrph  E  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y }
 ) ( <. V ,  E >. Neighbors  x )  =  {
 y ,  z }
 ) )
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