P. 164 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16301-16400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grstructd2dom 16301*	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.) (Revised by AV, 9-Jun-2021.)
|- ( 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 -> 2o ~<_ dom S )   &    |- ( ph -> ( Base ` S ) = V )   &    |- ( ph -> (.ef ` S ) = E )   =>    |- ( ph -> [. S / g ]. ps )
Theorem	gropeld 16302*	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 )
Theorem	grstructeld2dom 16303*	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.) (Revised by AV, 9-Jun-2021.)
|- ( 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 -> 2o ~<_ dom S )   &    |- ( ph -> ( Base ` S ) = V )   &    |- ( ph -> (.ef ` S ) = E )   =>    |- ( ph -> S e. C )
Theorem	setsvtx 16304	The vertices of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 18-Jan-2020.) (Revised by AV, 16-Nov-2021.)
|- I = (.ef ` ndx )   &    |- ( ph -> G Struct X )   &    |- ( ph -> ( Base ` ndx ) e. dom G )   &    |- ( ph -> E e. W )   =>    |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = ( Base ` G ) )
Theorem	setsiedg 16305	The (indexed) edges of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
|- I = (.ef ` ndx )   &    |- ( ph -> G Struct X )   &    |- ( ph -> ( Base ` ndx ) e. dom G )   &    |- ( ph -> E e. W )   =>    |- ( ph -> (iEdg ` ( G sSet <. I , E >. ) ) = E )
12.1.2.4  Degenerated cases of representations of graphs
Theorem	vtxval0 16306	Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
|- (Vtx ` (/) ) = (/)
Theorem	iedgval0 16307	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 ` (/) ) = (/)
Theorem	vtxvalprc 16308	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 ) = (/) )
Theorem	iedgvalprc 16309	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 ) = (/) )
12.1.3  Edges as range of the edge function
Syntax	cedg 16310	Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).
class Edg
Definition	df-edg 16311	Define the class of edges of a graph, see also definition "E = E(G)" in section I.1 of [Bollobas] p. 1. This definition is very general: It defines edges of a class as the range of its edge function (which does not even need to be a function). Therefore, this definition could also be used for hypergraphs, pseudographs and multigraphs. In these cases, however, the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. In some cases, this is no problem, so theorems with Edg are meaningful nevertheless. Usually, however, this definition is used only for undirected simple (hyper-/pseudo-)graphs (with or without loops). (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
|- Edg = ( g e. _V |-> ran (iEdg ` g ) )
Theorem	edgvalg 16312	The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.)
|- ( G e. V -> (Edg ` G ) = ran (iEdg ` G ) )
Theorem	edgval 16313	The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.)
|- (Edg ` G ) = ran (iEdg ` G )
Theorem	iedgedgg 16314	An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.)
|- E = (iEdg ` G )   =>    |- ( ( G e. V /\ Fun E /\ I e. dom E ) -> ( E ` I ) e. (Edg ` G ) )
Theorem	edgopval 16315	The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
|- ( ( V e. W /\ E e. X ) -> (Edg ` <. V , E >. ) = ran E )
Theorem	edgov 16316	The edges of a graph represented as ordered pair, shown as operation value. Although a little less intuitive, this representation is often used because it is shorter than the representation as function value of a graph given as ordered pair, see edgopval 16315. The representation ran E for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 13-Oct-2020.)
|- ( ( V e. W /\ E e. X ) -> ( VEdg E ) = ran E )
Theorem	edgstruct 16317	The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.)
|- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }   =>    |- ( ( V e. W /\ E e. X ) -> (Edg ` G ) = ran E )
Theorem	edgiedgbg 16318*	A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. V /\ Fun I ) -> ( E e. (Edg ` G ) <-> E. x e. dom I E = ( I ` x ) ) )
Theorem	edg0iedg0g 16319	There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.)
|- I = (iEdg ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. V /\ Fun I ) -> ( E = (/) <-> I = (/) ) )
12.2  Undirected graphs
12.2.1  Undirected hypergraphs
Syntax	cuhgr 16320	Extend class notation with undirected hypergraphs.
class UHGraph
Syntax	cushgr 16321	Extend class notation with undirected simple hypergraphs.
class USHGraph
Definition	df-uhgrm 16322*	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 set of inhabited subsets of this set. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by Jim Kingdon, 29-Dec-2025.)
|- UHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } }
Definition	df-ushgrm 16323*	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 the 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 subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by Jim Kingdon, 31-Dec-2025.)
|- USHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } }
Theorem	isuhgrm 16324*	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 --> { s e. ~P V | E. j j e. s } ) )
Theorem	isushgrm 16325*	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-> { s e. ~P V | E. j j e. s } ) )
Theorem	uhgrfm 16326*	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 --> { s e. ~P V | E. j j e. s } )
Theorem	ushgrfm 16327*	The edge function of an undirected simple hypergraph is a one-to-one 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-> { s e. ~P V | E. j j e. s } )
Theorem	uhgrss 16328	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 )
Theorem	uhgreq12g 16329	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 ) )
Theorem	uhgrfun 16330	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 )
Theorem	uhgrm 16331*	An edge is an inhabited 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. j j e. ( E ` F ) )
Theorem	lpvtx 16332	The endpoints of a loop (which is an edge at index J) are two (identical) vertices A. (Contributed by AV, 1-Feb-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> A e. (Vtx ` G ) )
Theorem	ushgruhgr 16333	An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
|- ( G e. USHGraph -> G e. UHGraph )
Theorem	isuhgropm 16334*	The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 9-Oct-2020.)
|- ( ( V e. W /\ E e. X ) -> ( <. V , E >. e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
Theorem	uhgr0e 16335	The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
|- ( ph -> G e. W )   &    |- ( ph -> (iEdg ` G ) = (/) )   =>    |- ( ph -> G e. UHGraph )
Theorem	pw0ss 16336*	There are no inhabited subsets of the empty set. (Contributed by Jim Kingdon, 31-Dec-2025.)
|- { s e. ~P (/) | E. j j e. s } = (/)
Theorem	uhgr0vb 16337	The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.)
|- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph <-> (iEdg ` G ) = (/) ) )
Theorem	uhgr0 16338	The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
|- (/) e. UHGraph
Theorem	uhgrun 16339	The union U of two (undirected) hypergraphs G and H with the same vertex set V is a hypergraph with the vertex set V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
|- ( ph -> G e. UHGraph )   &    |- ( ph -> H e. UHGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   &    |- ( ph -> U e. W )   &    |- ( ph -> (Vtx ` U ) = V )   &    |- ( ph -> (iEdg ` U ) = ( E u. F ) )   =>    |- ( ph -> U e. UHGraph )
Theorem	uhgrunop 16340	The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If <. V , E >. and <. V , F >. are hypergraphs, then <. V , E u. F >. is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
|- ( ph -> G e. UHGraph )   &    |- ( ph -> H e. UHGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   =>    |- ( ph -> <. V , ( E u. F ) >. e. UHGraph )
Theorem	ushgrun 16341	The union U of two (undirected) simple hypergraphs G and H with the same vertex set V is a (not necessarily simple) hypergraph with the vertex set V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)
|- ( ph -> G e. USHGraph )   &    |- ( ph -> H e. USHGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   &    |- ( ph -> U e. W )   &    |- ( ph -> (Vtx ` U ) = V )   &    |- ( ph -> (iEdg ` U ) = ( E u. F ) )   =>    |- ( ph -> U e. UHGraph )
Theorem	ushgrunop 16342	The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If <. V , E >. and <. V , F >. are simple hypergraphs, then <. V , E u. F >. is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)
|- ( ph -> G e. USHGraph )   &    |- ( ph -> H e. USHGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   =>    |- ( ph -> <. V , ( E u. F ) >. e. UHGraph )
Theorem	incistruhgr 16343*	An incidence structure <. P , L , I >. "where P is a set whose elements are called points, L is a distinct set whose elements are called lines and I C_ ( P X. L ) is the incidence relation" (see Wikipedia "Incidence structure" (24-Oct-2020), https://en.wikipedia.org/wiki/Incidence_structure) implies an undirected hypergraph, if the incidence relation is right-total (to exclude empty edges). The points become the vertices, and the edge function is derived from the incidence relation by mapping each line ("edge") to the set of vertices incident to the line/edge. With P = ( Base ` S ) and by defining two new slots for lines and incidence relations and enhancing the definition of iEdg accordingly, it would even be possible to express that a corresponding incidence structure is an undirected hypergraph. By choosing the incident relation appropriately, other kinds of undirected graphs (pseudographs, multigraphs, simple graphs, etc.) could be defined. (Contributed by AV, 24-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) -> ( ( V = P /\ E = ( e e. L |-> { v e. P | v I e } ) ) -> G e. UHGraph ) )
12.2.2  Undirected pseudographs and multigraphs
Syntax	cupgr 16344	Extend class notation with undirected pseudographs.
class UPGraph
Syntax	cumgr 16345	Extend class notation with undirected multigraphs.
class UMGraph
Definition	df-upgren 16346*	Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set v (of "vertices") and a function e (representing indexed "edges") 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. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgren 16347). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
|- UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
Definition	df-umgren 16347*	Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set v (of "vertices") and a function e (representing indexed "edges") into subsets of v of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." (Contributed by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
|- UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } }
Theorem	isupgren 16348*	The property of being an undirected pseudograph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. U -> ( G e. UPGraph <-> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
Theorem	wrdupgren 16349*	The property of being an undirected pseudograph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. U /\ E e. Word X ) -> ( G e. UPGraph <-> E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
Theorem	upgrfen 16350*	The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfnen 16351 without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. UPGraph -> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
Theorem	upgrfnen 16351*	The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
Theorem	upgrss 16352	An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 29-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) C_ V )
Theorem	upgrm 16353*	An edge is an inhabited subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. j j e. ( E ` F ) )
Theorem	upgr1or2 16354	An edge of an undirected pseudograph has one or two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) )
Theorem	upgrfi 16355	An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. Fin )
Theorem	upgrex 16356*	An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x e. V E. y e. V ( E ` F ) = { x , y } )
Theorem	upgrop 16357	A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.)
|- ( G e. UPGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph )
Theorem	isumgren 16358*	The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )
Theorem	wrdumgren 16359*	The property of being an undirected multigraph, expressing the edges as "words". (Contributed by AV, 24-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. U /\ E e. Word X ) -> ( G e. UMGraph <-> E e. Word { x e. ~P V | x ~~ 2o } ) )
Theorem	umgrfen 16360*	The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfnen 16361 without explicitly specified domain of the edge function. (Contributed by AV, 24-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. UMGraph -> E : dom E --> { x e. ~P V | x ~~ 2o } )
Theorem	umgrfnen 16361*	The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by AV, 24-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UMGraph /\ E Fn A ) -> E : A --> { x e. ~P V | x ~~ 2o } )
Theorem	umgredg2en 16362	An edge of a multigraph has exactly two ends. (Contributed by AV, 24-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( ( G e. UMGraph /\ X e. dom E ) -> ( E ` X ) ~~ 2o )
Theorem	umgrbien 16363*	Show that an unordered pair is a valid edge in a multigraph. (Contributed by AV, 9-Mar-2021.)
|- X e. V   &    |- Y e. V   &    |- X =/= Y   =>    |- { X , Y } e. { x e. ~P V | x ~~ 2o }
Theorem	upgruhgr 16364	An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.)
|- ( G e. UPGraph -> G e. UHGraph )
Theorem	umgrupgr 16365	An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.)
|- ( G e. UMGraph -> G e. UPGraph )
Theorem	umgruhgr 16366	An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.)
|- ( G e. UMGraph -> G e. UHGraph )
Theorem	umgrnloopv 16367	In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 11-Dec-2020.)
|- E = (iEdg ` G )   =>    |- ( ( G e. UMGraph /\ M e. W ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )
Theorem	umgredgprv 16368	In a multigraph, an edge is an unordered pair of vertices. This theorem would not hold for arbitrary hyper-/pseudographs since either M or N could be proper classes ( ( E ` X ) would be a loop in this case), which are no vertices of course. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.)
|- E = (iEdg ` G )   &    |- V = (Vtx ` G )   =>    |- ( ( G e. UMGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )
Theorem	umgrnloop 16369*	In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.)
|- E = (iEdg ` G )   =>    |- ( G e. UMGraph -> ( E. x e. dom E ( E ` x ) = { M , N } -> M =/= N ) )
Theorem	umgrnloop0 16370*	A multigraph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 11-Dec-2020.)
|- E = (iEdg ` G )   =>    |- ( G e. UMGraph -> { x e. dom E | ( E ` x ) = { U } } = (/) )
Theorem	umgr0e 16371	The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
|- ( ph -> G e. W )   &    |- ( ph -> (iEdg ` G ) = (/) )   =>    |- ( ph -> G e. UMGraph )
Theorem	upgr0e 16372	The empty graph, with vertices but no edges, is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.)
|- ( ph -> G e. W )   &    |- ( ph -> (iEdg ` G ) = (/) )   =>    |- ( ph -> G e. UPGraph )
Theorem	upgr1elem1 16373*	Lemma for upgr1edc 16374. (Contributed by AV, 16-Oct-2020.) (Revised by Jim Kingdon, 6-Jan-2026.)
|- ( ph -> { B , C } e. S )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> DECID B = C )   =>    |- ( ph -> { { B , C } } C_ { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )
Theorem	upgr1edc 16374	A pseudograph with one edge. Such a graph is actually a simple pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
|- V = (Vtx ` G )   &    |- ( ph -> A e. X )   &    |- ( ph -> B e. V )   &    |- ( ph -> C e. V )   &    |- ( ph -> DECID B = C )   &    |- ( ph -> (iEdg ` G ) = { <. A , { B , C } >. } )   =>    |- ( ph -> G e. UPGraph )
Theorem	upgr0eop 16375	The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, and therefore also a multigraph ( G e. UMGraph). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.)
|- ( V e. W -> <. V , (/) >. e. UPGraph )
Theorem	upgr1eopdc 16376	A pseudograph with one edge. Such a graph is actually a simple pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
|- ( ph -> V e. W )   &    |- ( ph -> A e. X )   &    |- ( ph -> B e. V )   &    |- ( ph -> C e. V )   &    |- ( ph -> DECID B = C )   =>    |- ( ph -> <. V , { <. A , { B , C } >. } >. e. UPGraph )
Theorem	upgr1een 16377	A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16374 for a different way of specifying a graph with one edge. (Contributed by Jim Kingdon, 18-Mar-2026.)
|- ( ph -> K e. X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   =>    |- ( ph -> <. V , { <. K , E >. } >. e. UPGraph )
Theorem	umgr1een 16378	A graph with one non-loop edge is a multigraph. (Contributed by Jim Kingdon, 18-Mar-2026.)
|- ( ph -> K e. X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   =>    |- ( ph -> <. V , { <. K , E >. } >. e. UMGraph )
Theorem	upgrun 16379	The union U of two pseudographs G and H with the same vertex set V is a pseudograph with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)
|- ( ph -> G e. UPGraph )   &    |- ( ph -> H e. UPGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   &    |- ( ph -> U e. W )   &    |- ( ph -> (Vtx ` U ) = V )   &    |- ( ph -> (iEdg ` U ) = ( E u. F ) )   =>    |- ( ph -> U e. UPGraph )
Theorem	upgrunop 16380	The union of two pseudographs (with the same vertex set): If <. V , E >. and <. V , F >. are pseudographs, then <. V , E u. F >. is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)
|- ( ph -> G e. UPGraph )   &    |- ( ph -> H e. UPGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   =>    |- ( ph -> <. V , ( E u. F ) >. e. UPGraph )
Theorem	umgrun 16381	The union U of two multigraphs G and H with the same vertex set V is a multigraph with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.)
|- ( ph -> G e. UMGraph )   &    |- ( ph -> H e. UMGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   &    |- ( ph -> U e. W )   &    |- ( ph -> (Vtx ` U ) = V )   &    |- ( ph -> (iEdg ` U ) = ( E u. F ) )   =>    |- ( ph -> U e. UMGraph )
Theorem	umgrunop 16382	The union of two multigraphs (with the same vertex set): If <. V , E >. and <. V , F >. are multigraphs, then <. V , E u. F >. is a multigraph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
|- ( ph -> G e. UMGraph )   &    |- ( ph -> H e. UMGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   =>    |- ( ph -> <. V , ( E u. F ) >. e. UMGraph )
12.2.3  Loop-free graphs For a hypergraph, the property to be "loop-free" is expressed by I : dom I --> E with E = { x e. ~P V | 2o ~<_ x } and I = (iEdg ` G ). E is the set of edges which connect at least two vertices.
Theorem	umgrislfupgrenlem 16383	Lemma for umgrislfupgrdom 16384. (Contributed by AV, 27-Jan-2021.)
|- ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | x ~~ 2o }
Theorem	umgrislfupgrdom 16384*	A multigraph is a loop-free pseudograph. (Contributed by AV, 27-Jan-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( G e. UMGraph <-> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) )
Theorem	lfgredg2dom 16385*	An edge of a loop-free graph has at least two ends. (Contributed by AV, 23-Feb-2021.)
|- I = (iEdg ` G )   &    |- A = dom I   &    |- E = { x e. ~P V | 2o ~<_ x }   =>    |- ( ( I : A --> E /\ X e. A ) -> 2o ~<_ ( I ` X ) )
Theorem	lfgrnloopen 16386*	A loop-free graph has no loops. (Contributed by AV, 23-Feb-2021.)
|- I = (iEdg ` G )   &    |- A = dom I   &    |- E = { x e. ~P V | 2o ~<_ x }   =>    |- ( I : A --> E -> { x e. A | ( I ` x ) ~~ 1o } = (/) )
12.2.4  Edges as subsets of vertices of graphs
Theorem	uhgredgiedgb 16387*	In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
|- I = (iEdg ` G )   =>    |- ( G e. UHGraph -> ( E e. (Edg ` G ) <-> E. x e. dom I E = ( I ` x ) ) )
Theorem	uhgriedg0edg0 16388	A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.)
|- ( G e. UHGraph -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
Theorem	uhgredgm 16389*	An edge of a hypergraph is an inhabited subset of vertices. (Contributed by AV, 28-Nov-2020.)
|- ( ( G e. UHGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ E. x x e. E ) )
Theorem	edguhgr 16390	An edge of a hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.)
|- ( ( G e. UHGraph /\ E e. (Edg ` G ) ) -> E e. ~P (Vtx ` G ) )
Theorem	uhgredgrnv 16391	An edge of a hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 4-Jun-2021.)
|- ( ( G e. UHGraph /\ E e. (Edg ` G ) /\ N e. E ) -> N e. (Vtx ` G ) )
Theorem	upgredgssen 16392*	The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020.)
|- ( G e. UPGraph -> (Edg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
Theorem	umgredgssen 16393*	The set of edges of a multigraph is a subset of the set of proper unordered pairs of vertices. (Contributed by AV, 25-Nov-2020.)
|- ( G e. UMGraph -> (Edg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } )
Theorem	edgupgren 16394	Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)
|- ( ( G e. UPGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ ( E ~~ 1o \/ E ~~ 2o ) ) )
Theorem	edgumgren 16395	Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020.)
|- ( ( G e. UMGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ E ~~ 2o ) )
Theorem	uhgrvtxedgiedgb 16396*	In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 6-Jul-2022.)
|- I = (iEdg ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UHGraph /\ U e. V ) -> ( E. i e. dom I U e. ( I ` i ) <-> E. e e. E U e. e ) )
Theorem	upgredg 16397*	For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ C e. E ) -> E. a e. V E. b e. V C = { a , b } )
Theorem	umgredg 16398*	For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UMGraph /\ C e. E ) -> E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) )
Theorem	upgrpredgv 16399	An edge of a pseudograph always connects two vertices if the edge contains two sets. The two vertices/sets need not necessarily be different (loops are allowed). (Contributed by AV, 18-Nov-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )
Theorem	umgrpredgv 16400	An edge of a multigraph always connects two vertices. This theorem does not hold for arbitrary pseudographs: if either M or N is a proper class, then { M , N } e. E could still hold ( { M , N } would be either { M } or { N }, see prprc1 3821 or prprc2 3822, i.e. a loop), but M e. V or N e. V would not be true. (Contributed by AV, 27-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UMGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )