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Theorem List for Metamath Proof Explorer - 39201-39300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzoopth 39201 A half-open integer range can represent an ordered pair, analogous to fzopth 11870. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( ( M..^ N )  =  ( J..^ K )  <->  ( M  =  J  /\  N  =  K ) ) )
 
Theorem2ffzoeq 39202* Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N )
 --> Y ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `
  i )  =  ( P `  i
 ) ) ) )
 
Theoremfzosplitpr 39203 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  ( B  e.  ( ZZ>= `  A )  ->  ( A..^ ( B  +  2 ) )  =  ( ( A..^ B )  u.  { B ,  ( B  +  1
 ) } ) )
 
Theoremprinfzo0 39204 The intersection of a half-open integer range and the pair of its outer borders is empty. (Contributed by AV, 9-Jan-2021.)
 |-  ( M  e.  ZZ  ->  ( { M ,  N }  i^i  ( ( M  +  1 )..^ N ) )  =  (/) )
 
Theoremelfzr 39205 A member of a finite interval of integers is either a member of the corresponding half-open integer range or the upper bound of the interval. (Contributed by AV, 5-Feb-2021.)
 |-  ( K  e.  ( M ... N )  ->  ( K  e.  ( M..^ N )  \/  K  =  N ) )
 
Theoremelfzo0l 39206 A member of a half-open range of nonnegative integers is either 0 or a member of the corresponding half-open range of positive integers. (Contributed by AV, 5-Feb-2021.)
 |-  ( K  e.  ( 0..^ N )  ->  ( K  =  0  \/  K  e.  ( 1..^ N ) ) )
 
Theoremelfzlmr 39207 A member of a finite interval of integers is either its lower bound or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.)
 |-  ( K  e.  ( M ... N )  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N )  \/  K  =  N ) )
 
Theoremelfz0lmr 39208 A member of a finite interval of nonnegative integers is either 0 or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.)
 |-  ( K  e.  ( 0 ... N )  ->  ( K  =  0  \/  K  e.  ( 1..^ N )  \/  K  =  N ) )
 
21.33.7.19  The ` # ` (set size) function - extension
 
Theoremnfile 39209 The size of any infinite set is always greater than or equal to the the size of any set. (Contributed by AV, 13-Nov-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  -.  B  e.  Fin )  ->  ( # `  A )  <_  ( # `  B ) )
 
Theoremprprrab 39210 The set of proper pairs of elements of a given set expressed in two ways. (Contributed by AV, 24-Nov-2020.)
 |-  { x  e.  ( ~P A  \  { (/) } )  |  ( # `  x )  =  2 }  =  { x  e.  ~P A  |  ( # `  x )  =  2 }
 
Theoremhash1n0 39211 If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.)
 |-  (
 ( A  e.  V  /\  ( # `  A )  =  1 )  ->  A  =/=  (/) )
 
21.33.7.20  Extended nonnegative integers

The function values of the hash (set size) function are either nonnegative integers or positive infinity, see hashf 12560. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers  RR*, see df-xr 9710. The definition of extended nonnegative integers can be used in Ramsey theory, because the Ramsey number is either a nonnegative integer or plus infinity, see ramcl2 15028, or for the degree of polynomials, see mdegcl 23074, or for the degree of vertices in graph theory, see vdgrf 25682.

 
Syntaxcxnn0 39212 The set of extended nonnegative integers.
 class NN0*
 
Definitiondf-xnn0 39213 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers  RR*, see df-xr 9710. (Contributed by AV, 10-Dec-2020.)
 |- NN0*  =  (
 NN0  u.  { +oo }
 )
 
Theoremelxnn0 39214 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  <->  ( A  e.  NN0 
 \/  A  = +oo ) )
 
Theoremnn0ssxnn0 39215 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
 |-  NN0  C_ NN0*
 
Theoremnn0xnn0 39216 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  e. NN0* )
 
Theoremxnn0xr 39217 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  e.  RR* )
 
Theoremxnn0xrge0 39218 An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  e.  ( 0 [,] +oo ) )
 
Theorem0xnn0 39219 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  0  e. NN0*
 
Theorempnf0xnn0 39220 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |- +oo  e. NN0*
 
Theoremnn0nepnf 39221 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  =/= +oo )
 
Theoremnn0xnn0d 39222 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e. NN0* )
 
Theoremnn0nepnfd 39223 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  =/= +oo )
 
Theoremxnn0nemnf 39224 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  =/= -oo )
 
Theoremxnn0xrnemnf 39225 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  ( A  e.  RR*  /\  A  =/= -oo ) )
 
Theoremxnn0ge0 39226 An extended nonnegative integer is greater than or equal to 0, see also nn0pnfge0 11468. (Contributed by AV, 10-Dec-2020.) (Proof modification is discouraged.)
 |-  ( N  e. NN0*  ->  0  <_  N )
 
Theoremxnn0nnn0pnf 39227 An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  (
 ( N  e. NN0*  /\  -.  N  e.  NN0 )  ->  N  = +oo )
 
Theoremxnn0xaddcl 39228 The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.)
 |-  (
 ( A  e. NN0*  /\  B  e. NN0* )  ->  ( A +e B )  e. NN0*
 )
 
Theoremxnn0add4d 39229 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 11623. (Contributed by AV, 12-Dec-2020.)
 |-  ( ph  ->  A  e. NN0* )   &    |-  ( ph  ->  B  e. NN0* )   &    |-  ( ph  ->  C  e. NN0* )   &    |-  ( ph  ->  D  e. NN0* )   =>    |-  ( ph  ->  ( ( A +e B ) +e ( C +e D ) )  =  ( ( A +e C ) +e ( B +e D ) ) )
 
Theoremxnn0xadd0 39230 The sum of two extended nonnegative integers is  0 iff each of the two extended nonnegative integers is 
0. (Contributed by AV, 14-Dec-2020.)
 |-  (
 ( A  e. NN0*  /\  B  e. NN0* )  ->  ( ( A +e B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
 
Theoremhashfxnn0 39231 The size function is a function into the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
 |-  # : _V -->NN0*
 
Theoremhashxnn0 39232 The value of the hash function for a set is an extended nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Dec-2020.)
 |-  ( M  e.  V  ->  ( # `  M )  e. NN0*
 )
 
21.33.7.21  Finite and infinite sums - extension
 
Theoremfsummsndifre 39233* A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\ 
 A. k  e.  A  B  e.  ZZ )  -> 
 sum_ k  e.  ( A  \  { X }
 ) B  e.  RR )
 
Theoremfsumsplitsndif 39234* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\  X  e.  A  /\  A. k  e.  A  B  e.  ZZ )  ->  sum_ k  e.  A  B  =  (
 sum_ k  e.  ( A  \  { X }
 ) B  +  [_ X  /  k ]_ B ) )
 
Theoremfsummmodsndifre 39235* A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\  N  e.  NN  /\  A. k  e.  A  B  e.  ZZ )  ->  sum_ k  e.  ( A  \  { X } ) ( B 
 mod  N )  e.  RR )
 
Theoremfsummmodsnunz 39236* A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
 |-  (
 ( A  e.  Fin  /\  N  e.  NN  /\  A. k  e.  ( A  u.  { z }
 ) B  e.  ZZ )  ->  sum_ k  e.  ( A  u.  { z }
 ) ( B  mod  N )  e.  ZZ )
 
21.33.8  Graph theory (revised)
 
21.33.8.1  The edge function extractor for extensible structures
 
Syntaxcedgf 39237 Extend class notation with an edge function.
 class .ef
 
Definitiondf-edgf 39238 Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)
 |- .ef  = Slot ; 1 8
 
Theoremedgfndxnn 39239 The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.)
 |-  (.ef ` 
 ndx )  e.  NN
 
Theoremedgfndxid 39240 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  e.  V  ->  (.ef `  G )  =  ( G `  (.ef `  ndx ) ) )
 
Theorembaseltedgf 39241 The index value of the  Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.)
 |-  ( Base `  ndx )  < 
 (.ef `  ndx )
 
Theoremslotsbaseefdif 39242 The slots  Base and .ef are different. (Contributed by AV, 21-Sep-2020.)
 |-  ( Base `  ndx )  =/=  (.ef `  ndx )
 
21.33.8.2  Vertices and edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath:  <. V ,  E >.. Another way is to regard a graph as a mathematical structure, which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 15178.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 39245 and df-iedg 39246), which provide the vertices resp. (indexed) edges of an arbitrary class  G which represents a graph:  (Vtx `  G
) resp.  (iEdg `  G ). Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G)."). For example, e1 = e(1) = { a, b } and e2 = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e1 and e2 connecting the same two vertices a and b, we would have e(e1) = e(e2) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair ( G  e.  ( _V  X.  _V )), the set of vertices is  (Vtx `  G
)  =  ( 1st `  G ) (see opvtxval 39249) and the set of (indexed) edges is  (iEdg `  G
)  =  ( 2nd `  G ) (see opiedgval 39252), or if  G is given as ordered pair  G  =  <. V ,  E >., the set of vertices is  (Vtx `  G
)  =  V (see opvtxfv 39250) and the set of (indexed) edges is  (iEdg `  G
)  =  E (see opiedgfv 39253).

And for a graph represented as extensible structure ( G Struct  <. ( Base `  ndx ) ,  (.ef `  ndx ) >.), the set of vertices is  (Vtx `  G
)  =  ( Base `  G ) (see funvtxval 39262) and the set of (indexed) edges is  (iEdg `  G
)  =  (.ef `  G ) (see funiedgval 39263), or if  G is given in its simplest form as extensible structure with two slots ( G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  E >. }), the set of vertices is  (Vtx `  G )  =  V (see struct2grvtx 39271) and the set of (indexed) edges is  (iEdg `  G )  =  E (see struct2griedg 39272).

These two representations are convertible, see graop 39273 and grastruct 39274: If  G is a graph (for example  G  =  <. V ,  E >.), then  H  =  { <. ( Base `  ndx ) ,  (Vtx `  G
) >. ,  <. (.ef `  ndx ) ,  (iEdg `  G ) >. } represents essentially the same graph, and if  G is a graph (for example  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  E >. }), then  H  =  <. (Vtx
`  G ) ,  (iEdg `  G ) >. represents essentially the same graph. In both cases,  (Vtx `  G )  =  (Vtx `  H ) and  (iEdg `  G
)  =  (iEdg `  H ) hold. Theorems gropd 39275 and gropeld 39277 show that if any representation of a graph with vertices  V and edges  E has a certain property, then the ordered pair  <. V ,  E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices  V and edges  E) has this property. Analogously, theorems grstructd 39276 and grstructeld 39278 show that if any representation of a graph with vertices  V and edges  E has a certain property, then any extensible structure with base set  V and value  E in the slot for edge functions (which is also such a representation of a graph with vertices  V and edges  E) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be  G  =  <. V ,  (/) >. or  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  (/) >. }, a structure without a slot for edges can be used:  G  =  { <. ( Base `  ndx ) ,  V >. }, see snstrvtxval 39279 and snstriedgval 39280. Analogously, the empty set 
(/) can be used to represent the null graph, see vtxval0 39281 and iedgval0 39282, which can also be represented by  G  =  <.
(/) ,  (/) >. or  G  =  { <. ( Base `  ndx ) ,  (/) >. ,  <. (.ef
`  ndx ) ,  (/) >. }. Even proper classes can be used to represent the null graph, see vtxvalprc 39287 and iedgvalprc 39288.

Other classes should not be used to represent graphs, because there could be a degenerated behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 39283 resp. iedgvalsnop 39284, and vtxval3sn 39285 resp. iedgval3sn 39286.

 
Syntaxcvtx 39243 Extend class notation with the vertices of "graphs".
 class Vtx
 
Syntaxciedg 39244 Extend class notation with the indexed edges of "graphs".
 class iEdg
 
Definitiondf-vtx 39245 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, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)
 |- Vtx  =  ( g  e.  _V  |->  if ( g  e.  ( _V  X.  _V ) ,  ( 1st `  g
 ) ,  ( Base `  g ) ) )
 
Definitiondf-iedg 39246 Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)
 |- iEdg  =  ( g  e.  _V  |->  if ( g  e.  ( _V  X.  _V ) ,  ( 2nd `  g
 ) ,  (.ef `  g ) ) )
 
Theoremvtxval 39247 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
 |-  ( G  e.  V  ->  (Vtx `  G )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 1st `  G ) ,  ( Base `  G ) ) )
 
Theoremiedgval 39248 The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  e.  V  ->  (iEdg `  G )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 2nd `  G ) ,  (.ef `  G ) ) )
 
Theoremopvtxval 39249 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
 |-  ( G  e.  ( _V  X. 
 _V )  ->  (Vtx `  G )  =  ( 1st `  G )
 )
 
Theoremopvtxfv 39250 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  (Vtx `  <. V ,  E >. )  =  V )
 
Theoremopvtxov 39251 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( VVtx E )  =  V )
 
Theoremopiedgval 39252 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  e.  ( _V  X. 
 _V )  ->  (iEdg `  G )  =  ( 2nd `  G )
 )
 
Theoremopiedgfv 39253 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  (iEdg `  <. V ,  E >. )  =  E )
 
Theoremopiedgov 39254 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( ViEdg E )  =  E )
 
Theoremfunvtxdm2val 39255 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.)
 |-  A  e.  W   &    |-  B  e.  Z   =>    |-  (
 ( ( G  e.  V  /\  Fun  G )  /\  A  =/=  B  /\  { A ,  B }  C_ 
 dom  G )  ->  (Vtx `  G )  =  (
 Base `  G ) )
 
Theoremfuniedgdm2val 39256 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.)
 |-  A  e.  W   &    |-  B  e.  Z   =>    |-  (
 ( ( G  e.  V  /\  Fun  G )  /\  A  =/=  B  /\  { A ,  B }  C_ 
 dom  G )  ->  (iEdg `  G )  =  (.ef `  G ) )
 
Theoremfunvtxval0 39257 The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.)
 |-  S  e.  Z   =>    |-  ( ( ( G  e.  V  /\  Fun  G )  /\  S  =/=  ( Base `  ndx )  /\  { ( Base `  ndx ) ,  S }  C_  dom  G )  ->  (Vtx `  G )  =  ( Base `  G ) )
 
Theoremfunvtxdmge2val 39258 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  (Vtx `  G )  =  (
 Base `  G ) )
 
Theoremfuniedgdmge2val 39259 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  (iEdg `  G )  =  (.ef `  G ) )
 
Theorembasvtxval 39260 The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  2  <_  ( # `  dom  G ) )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  <. ( Base `  ndx ) ,  V >.  e.  G )   =>    |-  ( ph  ->  (Vtx `  G )  =  V )
 
Theoremedgfiedgval 39261 The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  2  <_  ( # `  dom  G ) )   &    |-  ( ph  ->  E  e.  Y )   &    |-  ( ph  ->  <. (.ef `  ndx ) ,  E >.  e.  G )   =>    |-  ( ph  ->  (iEdg `  G )  =  E )
 
Theoremfunvtxval 39262 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  {
 ( Base `  ndx ) ,  (.ef `  ndx ) }  C_ 
 dom  G )  ->  (Vtx `  G )  =  (
 Base `  G ) )
 
Theoremfuniedgval 39263 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  {
 ( Base `  ndx ) ,  (.ef `  ndx ) }  C_ 
 dom  G )  ->  (iEdg `  G )  =  (.ef `  G ) )
 
Theoremstructvtxvallem 39264 Lemma for structvtxval 39265 and structiedg0val 39266. (Contributed by AV, 23-Sep-2020.)
 |-  S  e.  NN   &    |-  ( Base `  ndx )  <  S   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( G  e.  _V 
 /\  Fun  G  /\  { ( Base `  ndx ) ,  S }  C_  dom  G ) )
 
Theoremstructvtxval 39265 The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.)
 |-  S  e.  NN   &    |-  ( Base `  ndx )  <  S   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  (Vtx `  G )  =  V )
 
Theoremstructiedg0val 39266 The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.)
 |-  S  e.  NN   &    |-  ( Base `  ndx )  <  S   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  (iEdg `  G )  =  (/) )
 
Theoremstructgrssvtxlem 39267 Lemma for structgrssvtx 39268 and structgrssiedg 39269. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  E  e.  Z )   &    |-  ( ph  ->  { <. ( Base ` 
 ndx ) ,  V >. ,  <. (.ef `  ndx ) ,  E >. } 
 C_  G )   =>    |-  ( ph  ->  2 
 <_  ( # `  dom  G ) )
 
Theoremstructgrssvtx 39268 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  E  e.  Z )   &    |-  ( ph  ->  { <. ( Base ` 
 ndx ) ,  V >. ,  <. (.ef `  ndx ) ,  E >. } 
 C_  G )   =>    |-  ( ph  ->  (Vtx `  G )  =  V )
 
Theoremstructgrssiedg 39269 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  E  e.  Z )   &    |-  ( ph  ->  { <. ( Base ` 
 ndx ) ,  V >. ,  <. (.ef `  ndx ) ,  E >. } 
 C_  G )   =>    |-  ( ph  ->  (iEdg `  G )  =  E )
 
Theoremstruct2grstr 39270 A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.)
 |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
 <. (.ef `  ndx ) ,  E >. }   =>    |-  G Struct  <. ( Base `  ndx ) ,  (.ef `  ndx ) >.
 
Theoremstruct2grvtx 39271 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.)
 |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
 <. (.ef `  ndx ) ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  (Vtx `  G )  =  V )
 
Theoremstruct2griedg 39272 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 23-Sep-2020.)
 |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
 <. (.ef `  ndx ) ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  (iEdg `  G )  =  E )
 
Theoremgraop 39273 Any representation of a graph  G (especially as extensible structure  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  E >. }) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)
 |-  H  =  <. (Vtx `  G ) ,  (iEdg `  G ) >.   =>    |-  ( (Vtx `  G )  =  (Vtx `  H )  /\  (iEdg `  G )  =  (iEdg `  H ) )
 
Theoremgrastruct 39274 Any representation of a graph  G (especially as ordered pair  G  =  <. V ,  E >.) is convertible in a representation of the graph as extensible structure. (Contributed by AV, 8-Oct-2020.)
 |-  H  =  { <. ( Base `  ndx ) ,  (Vtx `  G ) >. ,  <. (.ef `  ndx ) ,  (iEdg `  G ) >. }   =>    |-  ( (Vtx `  G )  =  (Vtx `  H )  /\  (iEdg `  G )  =  (iEdg `  H ) )
 
Theoremgropd 39275* If any representation of a graph with vertices  V and edges  E has a certain property  ps, then the ordered pair  <. V ,  E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices  V and edges  E) has this property. (Contributed by AV, 11-Oct-2020.)
 |-  ( ph  ->  A. g ( ( (Vtx `  g )  =  V  /\  (iEdg `  g )  =  E )  ->  ps ) )   &    |-  ( ph  ->  V  e.  U )   &    |-  ( ph  ->  E  e.  W )   =>    |-  ( ph  ->  [. <. V ,  E >.  /  g ]. ps )
 
Theoremgrstructd 39276* If any representation of a graph with vertices  V and edges  E has a certain property  ps, then any structure with base set  V and value  E in the slot for edge functions (which is such a representation of a graph with vertices  V and edges  E) has this property. (Contributed by AV, 12-Oct-2020.)
 |-  ( ph  ->  A. g ( ( (Vtx `  g )  =  V  /\  (iEdg `  g )  =  E )  ->  ps ) )   &    |-  ( ph  ->  V  e.  U )   &    |-  ( ph  ->  E  e.  W )   &    |-  ( ph  ->  S  e.  X )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  2  <_  ( # `
  dom  S )
 )   &    |-  ( ph  ->  ( Base `  S )  =  V )   &    |-  ( ph  ->  (.ef `  S )  =  E )   =>    |-  ( ph  ->  [. S  /  g ]. ps )
 
Theoremgropeld 39277* If any representation of a graph with vertices  V and edges  E is an element of an arbitrary class  C, then the ordered pair  <. V ,  E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices  V and edges  E) is an element of this class 
C. (Contributed by AV, 11-Oct-2020.)
 |-  ( ph  ->  A. g ( ( (Vtx `  g )  =  V  /\  (iEdg `  g )  =  E )  ->  g  e.  C ) )   &    |-  ( ph  ->  V  e.  U )   &    |-  ( ph  ->  E  e.  W )   =>    |-  ( ph  ->  <. V ,  E >.  e.  C )
 
Theoremgrstructeld 39278* If any representation of a graph with vertices  V and edges  E is an element of an arbitrary class  C, then any structure with base set  V and value  E in the slot for edge functions (which is such a representation of a graph with vertices  V and edges  E) is an element of this class  C. (Contributed by AV, 12-Oct-2020.)
 |-  ( ph  ->  A. g ( ( (Vtx `  g )  =  V  /\  (iEdg `  g )  =  E )  ->  g  e.  C ) )   &    |-  ( ph  ->  V  e.  U )   &    |-  ( ph  ->  E  e.  W )   &    |-  ( ph  ->  S  e.  X )   &    |-  ( ph  ->  Fun 
 S )   &    |-  ( ph  ->  2 
 <_  ( # `  dom  S ) )   &    |-  ( ph  ->  (
 Base `  S )  =  V )   &    |-  ( ph  ->  (.ef `  S )  =  E )   =>    |-  ( ph  ->  S  e.  C )
 
Theoremsnstrvtxval 39279 The set of vertices of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See vtxvalsnop 39283 for the (degenerated) case where  V  =  (
Base `  ndx ). (Contributed by AV, 23-Sep-2020.)
 |-  V  e.  X   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. }   =>    |-  ( V  =/=  ( Base `  ndx )  ->  (Vtx `  G )  =  V )
 
Theoremsnstriedgval 39280 The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 39284 for the (degenerated) case where  V  =  (
Base `  ndx ). (Contributed by AV, 24-Sep-2020.)
 |-  V  e.  X   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. }   =>    |-  ( V  =/=  ( Base `  ndx )  ->  (iEdg `  G )  =  (/) )
 
Theoremvtxval0 39281 Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
 |-  (Vtx `  (/) )  =  (/)
 
Theoremiedgval0 39282 Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
 |-  (iEdg `  (/) )  =  (/)
 
Theoremvtxvalsnop 39283 Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.)
 |-  B  e.  V   &    |-  G  =  { <. B ,  B >. }   =>    |-  (Vtx `  G )  =  { B }
 
Theoremiedgvalsnop 39284 Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.)
 |-  B  e.  V   &    |-  G  =  { <. B ,  B >. }   =>    |-  (iEdg `  G )  =  { B }
 
Theoremvtxval3sn 39285 Degenerated case 3 for vertices: The set of vertices of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   =>    |-  (Vtx `  { { { A } } } )  =  { A }
 
Theoremiedgval3sn 39286 Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   =>    |-  (iEdg `  { { { A } } } )  =  { A }
 
Theoremvtxvalprc 39287 Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
 |-  ( C  e/  _V  ->  (Vtx `  C )  =  (/) )
 
Theoremiedgvalprc 39288 Degenerated case 4 for edges: The set of indexed edges of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
 |-  ( C  e/  _V  ->  (iEdg `  C )  =  (/) )
 
21.33.8.3  Undirected hypergraphs
 
Syntaxcuhgr 39289 Extend class notation with undirected hypergraphs.
 class UHGraph
 
Syntaxcushgr 39290 Extend class notation with undirected simple hypergraphs.
 class USHGraph
 
Definitiondf-uhgr 39291* Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 
v (of "vertices") and a function  e (representing indexed "edges") into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 8-Oct-2020.)
 |- UHGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
 )  /  e ]. e : dom  e --> ( ~P v  \  { (/) } ) }
 
Definitiondf-ushgr 39292* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function  e is an injective (one-to-one) function into subsets of the set of vertices  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 are non-empty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by AV, 8-Oct-2020.)
 |- USHGraph  =  {
 g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
 )  /  e ]. e : dom  e -1-1-> ( ~P v  \  { (/) } ) }
 
Theoremisuhgr 39293 The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e.  U  ->  ( G  e. UHGraph  <->  E : dom  E --> ( ~P V  \  { (/)
 } ) ) )
 
Theoremisushgr 39294 The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e.  U  ->  ( G  e. USHGraph  <->  E : dom  E -1-1-> ( ~P V  \  { (/)
 } ) ) )
 
Theoremuhgrf 39295 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, 9-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UHGraph  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
 
Theoremushgrf 39296 The edge function of an undirected simple hypergraph is a function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USHGraph  ->  E : dom  E -1-1-> ( ~P V  \  { (/) } ) )
 
Theoremuhgrss 39297 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  F  e.  dom  E )  ->  ( E `  F ) 
 C_  V )
 
Theoremuhgreq12g 39298 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  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  W  =  (Vtx `  H )   &    |-  F  =  (iEdg `  H )   =>    |-  (
 ( ( G  e.  X  /\  H  e.  Y )  /\  ( V  =  W  /\  E  =  F ) )  ->  ( G  e. UHGraph 
 <->  H  e. UHGraph  ) )
 
Theoremuhgrfun 39299 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 15-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UHGraph  ->  Fun  E )
 
Theoremuhgrn0 39300 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.)
 |-  E  =  (iEdg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `
  F )  =/=  (/) )
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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 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