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Type | Label | Description |
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Statement | ||
Theorem | fzoopth 39201 | A half-open integer range can represent an ordered pair, analogous to fzopth 11870. (Contributed by Alexander van der Vekens, 1-Jul-2018.) |
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Theorem | 2ffzoeq 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.) |
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Theorem | fzosplitpr 39203 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
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Theorem | prinfzo0 39204 | The intersection of a half-open integer range and the pair of its outer borders is empty. (Contributed by AV, 9-Jan-2021.) |
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Theorem | elfzr 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.) |
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Theorem | elfzo0l 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.) |
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Theorem | elfzlmr 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.) |
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Theorem | elfz0lmr 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.) |
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Theorem | nfile 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.) |
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Theorem | prprrab 39210 | The set of proper pairs of elements of a given set expressed in two ways. (Contributed by AV, 24-Nov-2020.) |
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Theorem | hash1n0 39211 | If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.) |
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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 | ||
Syntax | cxnn0 39212 | The set of extended nonnegative integers. |
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Definition | df-xnn0 39213 |
Define the set of extended nonnegative integers that includes positive
infinity. Analogue of the extension of the real numbers ![]() |
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Theorem | elxnn0 39214 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0ssxnn0 39215 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0xnn0 39216 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0xr 39217 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0xrge0 39218 | An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.) |
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Theorem | 0xnn0 39219 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
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Theorem | pnf0xnn0 39220 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0nepnf 39221 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0xnn0d 39222 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0nepnfd 39223 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0nemnf 39224 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0xrnemnf 39225 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0ge0 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.) |
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Theorem | xnn0nnn0pnf 39227 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0xaddcl 39228 | The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0add4d 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.) |
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Theorem | xnn0xadd0 39230 |
The sum of two extended nonnegative integers is ![]() ![]() |
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Theorem | hashfxnn0 39231 | The size function is a function into the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
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Theorem | hashxnn0 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.) |
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Theorem | fsummsndifre 39233* | A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.) |
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Theorem | fsumsplitsndif 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.) |
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Theorem | fsummmodsndifre 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.) |
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Theorem | fsummmodsnunz 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.) |
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Syntax | cedgf 39237 | Extend class notation with an edge function. |
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Definition | df-edgf 39238 | Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.) |
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Theorem | edgfndxnn 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.) |
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Theorem | edgfndxid 39240 | The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) |
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Theorem | baseltedgf 39241 |
The index value of the ![]() |
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Theorem | slotsbaseefdif 39242 |
The slots ![]() |
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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:
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
The result of these functions are as expected: for a graph represented as
ordered pair (
And for a graph represented as extensible structure
(
These two representations are convertible, see graop 39273 and grastruct 39274:
If
Besides the usual way to represent graphs without edges (consisting of
unconnected vertices only), which would be 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. | ||
Syntax | cvtx 39243 | Extend class notation with the vertices of "graphs". |
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Syntax | ciedg 39244 | Extend class notation with the indexed edges of "graphs". |
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Definition | df-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.) |
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Definition | df-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.) |
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Theorem | vtxval 39247 | The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.) |
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Theorem | iedgval 39248 | The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.) |
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Theorem | opvtxval 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.) |
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Theorem | opvtxfv 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.) |
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Theorem | opvtxov 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.) |
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Theorem | opiedgval 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.) |
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Theorem | opiedgfv 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.) |
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Theorem | opiedgov 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.) |
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Theorem | funvtxdm2val 39255 | The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) |
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Theorem | funiedgdm2val 39256 | The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) |
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Theorem | funvtxval0 39257 | The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) |
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Theorem | funvtxdmge2val 39258 | The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) |
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Theorem | funiedgdmge2val 39259 | The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) |
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Theorem | basvtxval 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.) |
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Theorem | edgfiedgval 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.) |
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Theorem | funvtxval 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.) |
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Theorem | funiedgval 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.) |
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Theorem | structvtxvallem 39264 | Lemma for structvtxval 39265 and structiedg0val 39266. (Contributed by AV, 23-Sep-2020.) |
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Theorem | structvtxval 39265 | The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) |
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Theorem | structiedg0val 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.) |
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Theorem | structgrssvtxlem 39267 | Lemma for structgrssvtx 39268 and structgrssiedg 39269. (Contributed by AV, 14-Oct-2020.) |
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Theorem | structgrssvtx 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.) |
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Theorem | structgrssiedg 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.) |
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Theorem | struct2grstr 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.) |
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Theorem | struct2grvtx 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.) |
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Theorem | struct2griedg 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | graop 39273 |
Any representation of a graph ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | grastruct 39274 |
Any representation of a graph ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | gropd 39275* |
If any representation of a graph with vertices ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | grstructd 39276* |
If any representation of a graph with vertices ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | gropeld 39277* |
If any representation of a graph with vertices ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | grstructeld 39278* |
If any representation of a graph with vertices ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | snstrvtxval 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
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | snstriedgval 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
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vtxval0 39281 | Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | iedgval0 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vtxvalsnop 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | iedgvalsnop 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vtxval3sn 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | iedgval3sn 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vtxvalprc 39287 | Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | iedgvalprc 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Syntax | cuhgr 39289 | Extend class notation with undirected hypergraphs. |
![]() | ||
Syntax | cushgr 39290 | Extend class notation with undirected simple hypergraphs. |
![]() | ||
Definition | df-uhgr 39291* |
Define the class of all undirected hypergraphs. An undirected
hypergraph consists of a set ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Definition | df-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 ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | isuhgr 39293 | The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | isushgr 39294 | The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | uhgrf 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ushgrf 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | uhgrss 39297 | An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | uhgreq12g 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | uhgrfun 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | uhgrn0 39300 | An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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