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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuhgr0e 40401 The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ UHGraph )

Theoremuhgr0vb 40402 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.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))

Theoremuhgr0 40403 The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
∅ ∈ UHGraph

Theoremuhgrun 40404 The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.)
(𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph )

Theoremuhgrunop 40405 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, 𝐸𝐹 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.)
(𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph )

Theoremushgrun 40406 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 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
(𝜑𝐺 ∈ USHGraph )    &   (𝜑𝐻 ∈ USHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph )

Theoremushgrunop 40407 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, 𝐸𝐹 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.)
(𝜑𝐺 ∈ USHGraph )    &   (𝜑𝐻 ∈ USHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph )

Theoremuhgrstrrepelem 40408 Lemma for uhgrstrrepe 40409. (Contributed by AV, 7-Jun-2021.)
𝑉 = (Base‘𝐺)    &   𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct ⟨(Base‘ndx), 𝐼⟩)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐺𝑈)    &   (𝜑𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))    &   (𝜑𝐸𝑊)       (𝜑 → ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V ∧ Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet ⟨𝐼, 𝐸⟩)))

Theoremuhgrstrrepe 40409 Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑𝐺 Struct ⟨(Base‘ndx), 𝐼⟩), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) or only (𝜑 → Fun 𝐺). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.)
𝑉 = (Base‘𝐺)    &   𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct ⟨(Base‘ndx), 𝐼⟩)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐺𝑈)    &   (𝜑𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))    &   (𝜑𝐸𝑊)       (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph )

Theoremincistruhgr 40410* An incident 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 ⊆ (𝑃 × 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 (analogous to LineG and Itv) and enhancing the definition of iEdg accordingly, it would even be possible to express that a corresponding incident 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)       ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph ))

20.34.8.4  Undirected pseudographs and multigraphs

Syntaxcupgr 40411 Extend class notation with undirected pseudographs.
class UPGraph

Syntaxcumgr 40412 Extend class notation with undirected multigraphs.
class UMGraph

Definitiondf-upgr 40413* 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-umgr 40414). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.)
UPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Definitiondf-umgr 40414* 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." To provide uniform definitions for all kinds of graphs, 𝑥 ∈ (𝒫 𝑣 ∖ {∅}) is used as restriction of the class abstraction, although 𝑥 ∈ 𝒫 𝑣 would be sufficient (see prprrab 40300 and isumgrs 40426). (Contributed by AV, 24-Nov-2020.)
UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

Theoremisupgr 40415* 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)       (𝐺𝑈 → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremwrdupgr 40416* 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)       ((𝐺𝑈𝐸 ∈ Word 𝑋) → (𝐺 ∈ UPGraph ↔ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremupgrf 40417* The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 40418 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 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremupgrfn 40418* 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 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremupgrss 40419 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 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Theoremupgrn0 40420 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)

Theoremupgrle 40421 An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (#‘(𝐸𝐹)) ≤ 2)

Theoremupgrfi 40422 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 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ Fin)

Theoremupgrex 40423* 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 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})

Theoremupgrbi 40424* Show that an unordered pair is a valid edge in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 28-Feb-2021.)
𝑋𝑉    &   𝑌𝑉       {𝑋, 𝑌} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}

Theoremisumgr 40425* The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))

Theoremisumgrs 40426* The simplified property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremwrdumgr 40427* The property of being an undirected multigraph, expressing the edges as "words". (Contributed by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺𝑈𝐸 ∈ Word 𝑋) → (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremumgrf 40428* The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn 40429 without explicitly specified domain of the edge function. (Contributed by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UMGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremumgrfn 40429* 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 ∈ UMGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremumgredg2 40430 An edge of a multigraph has exactly two ends. (Contributed by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → (#‘(𝐸𝑋)) = 2)

Theoremumgrbi 40431* Show that an unordered pair is a valid edge in a multigraph. (Contributed by AV, 9-Mar-2021.)
𝑋𝑉    &   𝑌𝑉    &   𝑋𝑌       {𝑋, 𝑌} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}

Theoremupgruhgr 40432 An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.)
(𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )

Theoremumgrupgr 40433 An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph )

Theoremumgruhgr 40434 An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.)
(𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )

Theoremupgrle2 40435 An edge of an undirected pseudograph has at most two ends. (Contributed by AV, 6-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼𝑋)) ≤ 2)

Theoremumgrnloopv 40436 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 ∈ UMGraph ∧ 𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremumgredgprv 40437 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 ((𝐸𝑋) 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 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

Theoremumgrnloop 40438* 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 ∈ UMGraph → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremumgrnloop0 40439* A multigraph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UMGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremumgr0e 40440 The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ UMGraph )

Theoremupgr0e 40441 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.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ UPGraph )

Theoremupgr1elem 40442* Lemma for upgr1e 40443 and uspgr1e 40575. (Contributed by AV, 16-Oct-2020.)
(𝜑 → {𝐵, 𝐶} ∈ 𝑆)    &   (𝜑𝐵𝑊)       (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝑆 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremupgr1e 40443 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1e 40575. (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)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})       (𝜑𝐺 ∈ UPGraph )

Theoremupgr0eop 40444 The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, see usgr0eop 40577, and therefore also a multigraph (𝐺 ∈ UMGraph). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph )

Theoremupgr1eop 40445 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1eop 40578. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph )

Theoremupgr0eopALT 40446 Alternate proof of upgr0eop 40444, using the general theorem gropeld 40371 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 40444). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph )

Theoremupgr1eopALT 40447 Alternate proof of upgr1eop 40445, using the general theorem gropeld 40371 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr1eop 40445). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph )

Theoremupgrun 40448 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 (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.)
(𝜑𝐺 ∈ UPGraph )    &   (𝜑𝐻 ∈ UPGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UPGraph )

Theoremupgrunop 40449 The union of two pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are pseudographs, then 𝑉, 𝐸𝐹 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.)
(𝜑𝐺 ∈ UPGraph )    &   (𝜑𝐻 ∈ UPGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph )

Theoremumgrun 40450 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 (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.)
(𝜑𝐺 ∈ UMGraph )    &   (𝜑𝐻 ∈ UMGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UMGraph )

Theoremumgrunop 40451 The union of two multigraphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are multigraphs, then 𝑉, 𝐸𝐹 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.)
(𝜑𝐺 ∈ UMGraph )    &   (𝜑𝐻 ∈ UMGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UMGraph )

20.34.8.5  Loop-free graphs

For a hypergraph, the property to be "loop-free" is expressed by 𝐼:dom 𝐼𝐸 with 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} and 𝐼 = (iEdg‘𝐺). 𝐸 is the set of edges which connect at least two vertices.

Theoremumgrislfupgrlem 40452 Lemma for umgrislfupgr 40453 and usgrislfuspgr 40519. (Contributed by AV, 27-Jan-2021.)
({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}

Theoremumgrislfupgr 40453* A multigraph is a loop-free pseudograph. (Contributed by AV, 27-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UMGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}))

Theoremlfgredgge2 40454* 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 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}       ((𝐼:𝐴𝐸𝑋𝐴) → 2 ≤ (#‘(𝐼𝑋)))

Theoremlfgrnloop 40455* A loop-free graph has no loops. (Contributed by AV, 23-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}       (𝐼:𝐴𝐸 → {𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}} = ∅)

20.34.8.6  Edges as subsets of vertices of graphs

Syntaxcedga 40456 Extend class notation with the set of edges (of an undirected simple (pseudo)graph) Remark: If this definition (and all related theorems) are moved to main.set, the label should become "cedg".
class Edg

Definitiondf-edga 40457 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 even needs not 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 (e.g., edguhgr 40467). 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 ∈ V ↦ ran (iEdg‘𝑔))

Theoremedgaval 40458 The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
(𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Theoremedgaopval 40459 The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
((𝑉𝑊𝐸𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Theoremedgaov 40460 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 25607. 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.)
((𝑉𝑊𝐸𝑋) → (𝑉Edg𝐸) = ran 𝐸)

Theoremedgastruct 40461 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⟩}       ((𝑉𝑊𝐸𝑋) → (Edg‘𝐺) = ran 𝐸)

Theoremedgiedgb 40462* A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))

Theoremuhgredgiedgb 40463* In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))

Theoremedg0iedg0 40464 There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))

Theoremuhgriedg0edg0 40465 A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 15-Dec-2020.)
(𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))

Theoremuhgredgn0 40466 An edge of a hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Theoremedguhgr 40467 An edge of a hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺))

Theoremuhgredgrnv 40468 An edge of a hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 4-Jun-2021.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑁𝐸) → 𝑁 ∈ (Vtx‘𝐺))

Theoremuhgredgss 40469 The set of edges of a hypergraph is a subset of the powerset of vertices without the empty set. (Contributed by AV, 29-Nov-2020.)
(𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Theoremupgredgss 40470* 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 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremumgredgss 40471* The set of edges of a multigraph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ UMGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})

Theoremedgupgr 40472 Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)
((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (#‘𝐸) ≤ 2))

Theoremedgumgr 40473 Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020.)
((𝐺 ∈ UMGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐸) = 2))

Theoremuhgrvtxedgiedgb 40474* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))

Theoremupgredg 40475* For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})

Theoremumgredg 40476* 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 ∈ UMGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝐶 = {𝑎, 𝑏}))

Theoremumgrpredgav 40477 An edge of a multigraph always connects two vertices. Analogue of umgredgprv 40437. This theorem does not hold for arbitrary pseudographs: if either 𝑀 or 𝑁 is a proper class, then {𝑀, 𝑁} ∈ 𝐸 could still hold ({𝑀, 𝑁} would be either {𝑀} or {𝑁}, see prprc1 4146 or prprc2 4147, i.e. a loop), but 𝑀𝑉 or 𝑁𝑉 would not be true. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremupgredg2vtx 40478* For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 5-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐶𝐸𝐴𝐶) → ∃𝑏𝑉 𝐶 = {𝐴, 𝑏})

Theoremupgredgpr 40479 If a proper pair (of vertices) is a subset of an edge in a pseudograph, the pair is the edge. (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UPGraph ∧ 𝐶𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) ∧ (𝐴𝑈𝐵𝑊𝐴𝐵)) → {𝐴, 𝐵} = 𝐶)

Theoremumgredgne 40480 An edge of a multigraph always connects two different vertices. Analog of umgrnloopv 40436 resp. umgrnloop 40438. (Contributed by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → 𝑀𝑁)

Theoremumgrnloop2 40481 A multigraph has no loops. (Contributed by AV, 27-Oct-2020.) (Revised by AV, 30-Nov-2020.)
(𝐺 ∈ UMGraph → {𝑁, 𝑁} ∉ (Edg‘𝐺))

Theoremumgredgnlp 40482* An edge of a multigraph is not a loop. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝐶𝐸) → ¬ ∃𝑣 𝐶 = {𝑣})

20.34.8.7  Undirected simple graphs - basics

For undirected graphs, we will have the following hierarchy/taxonomy:

* Undirected Hypergraph: UHGraph

* Undirected simple Hypergraph: USHGraph => USHGraph ⊆ UHGraph (ushgruhgr 40396)

* Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 40432)

* Undirected loop-free hypergraph: ULFHGraph (not defined formally yet) => ULFHGraph ⊆ UHGraph

* Undirected loop-free simple hypergraph: ULFSHGraph (not defined formally yet) => ULFSHGraph ⊆ USHGraph and ULFSHGraph ULFHGraph

* Undirected simple Pseudograph: USPGraph => USPGraph ⊆ UPGraph (uspgrupgr 40511) and USPGraph ⊆ USHGraph (uspgrushgr 40510), see also uspgrupgrushgr 40512

* Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 40433) and UMGraph ⊆ ULFHGraph (umgrislfupgr 40453)

* Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 40513) and USGraph ⊆ UMGraph (usgrumgr 40514) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 40519) see also usgrumgruspgr 40515

In this section, "simple graph" will always stand for "undirected simple graph (without loops)" and "simple pseudograph" for "undirected simple pseudograph (which could have loops)".

Syntaxcuspgr 40483 Extend class notation with undirected simple pseudographs (which could have loops).
class USPGraph

Syntaxcusgr 40484 Extend class notation with undirected simple graphs (without loops).
class USGraph

Definitiondf-uspgr 40485* Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph (see uspgrupgr 40511) or a special undirected simple hypergraph (see uspgrushgr 40510), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) 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. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Definitiondf-usgr 40486* Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr 40513), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

Theoremisuspgr 40487* The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremisusgr 40488* The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))

Theoremuspgrf 40489* The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremusgrf 40490* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})

Theoremisusgrs 40491* The property of being a simple graph, simplified version of isusgr 40488. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) (Proof shortened by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremusgrfs 40492* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 40490. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremusgrfun 40493 The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.) (Revised by AV, 13-Oct-2020.)
(𝐺 ∈ USGraph → Fun (iEdg‘𝐺))

Theoremusgrusgra 40494 A simple graph represented by a class induces a representation as binary relation. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
(𝐺 ∈ USGraph → (Vtx‘𝐺) USGrph (iEdg‘𝐺))

Theoremusgredgss 40495* The set of edges of a simple graph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
(𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})

Theoremedgusgr 40496 An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐸) = 2))

Theoremisusgrop 40497* The property of being an undirected simple graph 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, 30-Nov-2020.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremusgrop 40498 A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.)
(𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph )

Theoremisausgr 40499* The property of an unordered pair to be an alternatively defined simple graph, defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremausgrusgrb 40500* The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝑉𝑋𝐸𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph ))

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