P. 166 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16501-16600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	struct2slots2dom 16501	There are at least two elements in an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.)
|- S e. NN   &    |- ( Base ` ndx ) < S   &    |- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> 2o ~<_ dom G )
Theorem	structvtxval 16502	The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- S e. NN   &    |- ( Base ` ndx ) < S   &    |- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> (Vtx ` G ) = V )
Theorem	structiedg0val 16503	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.) (Proof shortened by AV, 12-Nov-2021.)
|- S e. NN   &    |- ( Base ` ndx ) < S   &    |- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }   =>    |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (/) )
Theorem	structgr2slots2dom 16504	There are at least two elements in a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. Z )   &    |- ( ph -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } C_ G )   =>    |- ( ph -> 2o ~<_ dom G )
Theorem	structgrssvtx 16505	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.) (Proof shortened by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. Z )   &    |- ( ph -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } C_ G )   =>    |- ( ph -> (Vtx ` G ) = V )
Theorem	structgrssiedg 16506	The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. Z )   &    |- ( ph -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } C_ G )   =>    |- ( ph -> (iEdg ` G ) = E )
Theorem	struct2grstrg 16507	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 >. }   =>    |- ( ( V e. X /\ E e. Y ) -> G Struct <. ( Base ` ndx ) , (.ef ` ndx ) >. )
Theorem	struct2grvtx 16508	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 )
Theorem	struct2griedg 16509	The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> (iEdg ` G ) = E )
Theorem	gropd 16510*	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 )
Theorem	grstructd2dom 16511*	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 16512*	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 16513*	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 16514	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 16515	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 16516	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 16517	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 16518	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 16519	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 16520	Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).
class Edg
Definition	df-edg 16521	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 16522	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 16523	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 16524	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 16525	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 16526	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 16525. 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 16527	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 16528*	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 16529	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 16530	Extend class notation with undirected hypergraphs.
class UHGraph
Syntax	cushgr 16531	Extend class notation with undirected simple hypergraphs.
class USHGraph
Definition	df-uhgrm 16532*	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 16533*	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 16534*	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 16535*	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 16536*	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 16537*	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 16538	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 16539	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 16540	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 16541*	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 16542	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 16543	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 16544*	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 16545	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 16546*	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 16547	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 16548	The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
|- (/) e. UHGraph
Theorem	uhgrun 16549	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 16550	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 16551	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 16552	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 16553*	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 16554	Extend class notation with undirected pseudographs.
class UPGraph
Syntax	cumgr 16555	Extend class notation with undirected multigraphs.
class UMGraph
Definition	df-upgren 16556*	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 16557). (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 16557*	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 16558*	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 16559*	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 16560*	The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfnen 16561 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 16561*	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 16562	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 16563*	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 16564	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 16565	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 16566*	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 16567	A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.)
|- ( G e. UPGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph )
Theorem	isumgren 16568*	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 16569*	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 16570*	The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfnen 16571 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 16571*	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 16572	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 16573*	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 16574	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 16575	An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.)
|- ( G e. UMGraph -> G e. UPGraph )
Theorem	umgruhgr 16576	An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.)
|- ( G e. UMGraph -> G e. UHGraph )
Theorem	umgrnloopv 16577	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 16578	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 16579*	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 16580*	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 16581	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 16582	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 16583*	Lemma for upgr1edc 16584. (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 16584	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 16585	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 16586	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 16587	A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16584 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 16588	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 16589	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 16590	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 16591	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 16592	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 16593	Lemma for umgrislfupgrdom 16594. (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 16594*	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 16595*	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 16596*	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 16597*	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 16598	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 16599*	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 16600	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 ) )