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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstructgrssvtx 25701 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.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

Theoremstructgrssiedg 25702 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 14-Oct-2020.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

Theoremstruct2grstr 25703 A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩

Theoremstruct2grvtx 25704 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⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)

Theoremstruct2griedg 25705 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 23-Sep-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (iEdg‘𝐺) = 𝐸)

Theoremgraop 25706 Any representation of a graph 𝐺 (especially as extensible structure 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)
𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Theoremgrastruct 25707 Any representation of a graph 𝐺 (especially as ordered pair 𝐺 = ⟨𝑉, 𝐸) is convertible in a representation of the graph as extensible structure. (Contributed by AV, 8-Oct-2020.)
𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩}       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Theoremgropd 25708* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, 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.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)

Theoremgrstructd 25709* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, 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.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (#‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑[𝑆 / 𝑔]𝜓)

Theoremgropeld 25710* 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.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)

Theoremgrstructeld 25711* 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.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (#‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑𝑆𝐶)

16.1.2.4  Representations of graphs without edges

Theoremsnstrvtxval 25712 The set of vertices of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See vtxvalsnop 25716 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 23-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (Vtx‘𝐺) = 𝑉)

Theoremsnstriedgval 25713 The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 25717 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 24-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅)

16.1.2.5  Degenerated cases of representations of graphs

Theoremvtxval0 25714 Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(Vtx‘∅) = ∅

Theoremiedgval0 25715 Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(iEdg‘∅) = ∅

Theoremvtxvalsnop 25716 Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (Vtx‘𝐺) = {𝐵}

Theoremiedgvalsnop 25717 Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (iEdg‘𝐺) = {𝐵}

Theoremvtxval3sn 25718 Degenerated case 3 for vertices: The set of vertices of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V       (Vtx‘{{{𝐴}}}) = {𝐴}

Theoremiedgval3sn 25719 Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V       (iEdg‘{{{𝐴}}}) = {𝐴}

Theoremvtxvalprc 25720 Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
(𝐶 ∉ V → (Vtx‘𝐶) = ∅)

Theoremiedgvalprc 25721 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 ∉ V → (iEdg‘𝐶) = ∅)

16.1.3  Undirected hypergraphs

Syntaxcuhgr 25722 Extend class notation with undirected hypergraphs.
class UHGraph

Syntaxcushgr 25723 Extend class notation with undirected simple hypergraphs.
class USHGraph

Definitiondf-uhgr 25724* Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 8-Oct-2020.)
UHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}

Definitiondf-ushgr 25725* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subsets of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are non-empty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by AV, 8-Oct-2020.)
USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}

Theoremisuhgr 25726 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)       (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremisushgr 25727 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)       (𝐺𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))

Theoremuhgrf 25728 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 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))

Theoremushgrf 25729 The edge function of an undirected simple hypergraph is a function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USHGraph → 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))

Theoremuhgrss 25730 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 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Theoremuhgreq12g 25731 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)       (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph ))

Theoremuhgrfun 25732 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 ∈ UHGraph → Fun 𝐸)

Theoremuhgrn0 25733 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)

Theoremlpvtx 25734 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 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))

Theoremushgruhgr 25735 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
(𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph )

Theoremisuhgrop 25736 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.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremuhgr0e 25737 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 25738 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 25739 The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
∅ ∈ UHGraph

Theoremuhgrun 25740 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.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph )

Theoremuhgrunop 25741 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.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph )

Theoremushgrun 25742 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.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ USHGraph )    &   (𝜑𝐻 ∈ USHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph )

Theoremushgrunop 25743 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.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ USHGraph )    &   (𝜑𝐻 ∈ USHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph )

Theoremuhgrstrrepelem 25744 Lemma for uhgrstrrepe 25745. (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 25745 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 25746* 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 ))

16.1.4  Undirected pseudographs and multigraphs

Syntaxcupgr 25747 Extend class notation with undirected pseudographs.
class UPGraph

Syntaxcumgr 25748 Extend class notation with undirected multigraphs.
class UMGraph

Definitiondf-upgr 25749* 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 25750). (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 25750* 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 13112 and isumgrs 25762). (Contributed by AV, 24-Nov-2020.)
UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

Theoremisupgr 25751* 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 25752* 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 25753* The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 25754 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 25754* 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 25755 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 25756 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 25757 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 25758 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 25759* 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 25760* 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 25761* 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 25762* 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 25763* 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 25764* The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn 25765 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 25765* 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 25766 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 25767* 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 25768 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 25769 An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph )

Theoremumgruhgr 25770 An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.)
(𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )

Theoremupgrle2 25771 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 25772 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 25773 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 25774* 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 25775* 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 25776 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 25777 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 25778* Lemma for upgr1e 25779 and uspgr1e 40470. (Contributed by AV, 16-Oct-2020.)
(𝜑 → {𝐵, 𝐶} ∈ 𝑆)    &   (𝜑𝐵𝑊)       (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝑆 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremupgr1e 25779 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1e 40470. (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 25780 The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, see usgr0eop 40472, and therefore also a multigraph (𝐺 ∈ UMGraph). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph )

Theoremupgr1eop 25781 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1eop 40473. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph )

Theoremupgr0eopALT 25782 Alternate proof of upgr0eop 25780, using the general theorem gropeld 25710 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 25780). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph )

Theoremupgr1eopALT 25783 Alternate proof of upgr1eop 25781, using the general theorem gropeld 25710 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 25781). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph )

Theoremupgrun 25784 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.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UPGraph )    &   (𝜑𝐻 ∈ UPGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UPGraph )

Theoremupgrunop 25785 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.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UPGraph )    &   (𝜑𝐻 ∈ UPGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph )

Theoremumgrun 25786 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)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UMGraph )

Theoremumgrunop 25787 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)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UMGraph )

16.1.5  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 25788 Lemma for umgrislfupgr 25789 and usgrislfuspgr 40414. (Contributed by AV, 27-Jan-2021.)
({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}

Theoremumgrislfupgr 25789* 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 25790* 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 25791* 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}} = ∅)

16.1.6  Edges as subsets of vertices of graphs

Syntaxcedga 25792 Extend class notation with the set of edges (of an undirected simple (pseudo)graph) Remark: TODO-AV: If this definition (and all related theorems) are moved to main.set, the label should become "cedg".
class Edg

Definitiondf-edga 25793 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 25803). 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 25794 The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
(𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Theoremedgaopval 25795 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 25796 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 25869. 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 25797 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 25798* 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 25799* 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 25800 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 = ∅))

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