P. 264 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1vgrex 26301	A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
16.1.2.2  The vertices and edges of a graph represented as ordered pair
Theorem	opvtxval 26302	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺))
Theorem	opvtxfv 26303	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
Theorem	opvtxov 26304	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉Vtx𝐸) = 𝑉)
Theorem	opiedgval 26305	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd ‘𝐺))
Theorem	opiedgfv 26306	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
Theorem	opiedgov 26307	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉iEdg𝐸) = 𝐸)
Theorem	opvtxfvi 26308	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
⊢ 𝑉 ∈ V    &   ⊢ 𝐸 ∈ V    ⇒   ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉
Theorem	opiedgfvi 26309	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
⊢ 𝑉 ∈ V    &   ⊢ 𝐸 ∈ V    ⇒   ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸
16.1.2.3  The vertices and edges of a graph represented as extensible structure
Theorem	funvtxdmge2val 26310	The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))
Theorem	funiedgdmge2val 26311	The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))
Theorem	funvtxdm2val 26312	The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
Theorem	funiedgdm2val 26313	The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))
Theorem	funvtxval0 26314	The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
⊢ 𝑆 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
Theorem	basvtxval 26315	The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)    ⇒   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)
Theorem	edgfiedgval 26316	The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺))    &   ⊢ (𝜑 → 𝐸 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)    ⇒   ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸)
Theorem	funvtxval 26317	The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
⊢ ((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
Theorem	funiedgval 26318	The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
⊢ ((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))
Theorem	structvtxvallem 26319	Lemma for structvtxval 26320 and structiedg0val 26321. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.)
⊢ 𝑆 ∈ ℕ    &   ⊢ (Base‘ndx) < 𝑆    &   ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (♯‘dom 𝐺))
Theorem	structvtxval 26320	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 ∈ ℕ    &   ⊢ (Base‘ndx) < 𝑆    &   ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = 𝑉)
Theorem	structiedg0val 26321	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 ∈ ℕ    &   ⊢ (Base‘ndx) < 𝑆    &   ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅)
Theorem	structgrssvtxlem 26322	Lemma for structgrssvtx 26323 and structgrssiedg 26324. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺))
Theorem	structgrssvtx 26323	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.)
⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)
Theorem	structgrssiedg 26324	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.)
⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸)
Theorem	struct2grstr 26325	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)⟩
Theorem	struct2grvtx 26326	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 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = 𝑉)
Theorem	struct2griedg 26327	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 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = 𝐸)
Theorem	graop 26328	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))
Theorem	grastruct 26329	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))
Theorem	gropd 26330*	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))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)
Theorem	grstructd 26331*	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))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    &   ⊢ (𝜑 → Fun (𝑆 ∖ {∅}))    &   ⊢ (𝜑 → 2 ≤ (♯‘dom 𝑆))    &   ⊢ (𝜑 → (Base‘𝑆) = 𝑉)    &   ⊢ (𝜑 → (.ef‘𝑆) = 𝐸)    ⇒   ⊢ (𝜑 → [𝑆 / 𝑔]𝜓)
Theorem	gropeld 26332*	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) → 𝑔 ∈ 𝐶))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
Theorem	grstructeld 26333*	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) → 𝑔 ∈ 𝐶))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    &   ⊢ (𝜑 → Fun (𝑆 ∖ {∅}))    &   ⊢ (𝜑 → 2 ≤ (♯‘dom 𝑆))    &   ⊢ (𝜑 → (Base‘𝑆) = 𝑉)    &   ⊢ (𝜑 → (.ef‘𝑆) = 𝐸)    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐶)
Theorem	setsvtx 26334	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)    &   ⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Base‘𝐺))
Theorem	setsiedg 26335	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)    &   ⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸)
16.1.2.4  Representations of graphs without edges
Theorem	snstrvtxval 26336	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 26340 for the (degenerate) case where 𝑉 = (Base‘ndx). (Contributed by AV, 23-Sep-2020.)
⊢ 𝑉 ∈ V    &   ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩}    ⇒   ⊢ (𝑉 ≠ (Base‘ndx) → (Vtx‘𝐺) = 𝑉)
Theorem	snstriedgval 26337	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 26341 for the (degenerate) 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
Theorem	vtxval0 26338	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 26339	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	vtxvalsnop 26340	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.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐺 = {⟨𝐵, 𝐵⟩}    ⇒   ⊢ (Vtx‘𝐺) = {𝐵}
Theorem	iedgvalsnop 26341	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.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐺 = {⟨𝐵, 𝐵⟩}    ⇒   ⊢ (iEdg‘𝐺) = {𝐵}
Theorem	vtxval3sn 26342	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.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Vtx‘{{{𝐴}}}) = {𝐴}
Theorem	iedgval3sn 26343	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.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (iEdg‘{{{𝐴}}}) = {𝐴}
Theorem	vtxvalprc 26344	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) = ∅)
Theorem	iedgvalprc 26345	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  Edges as range of the edge function
Syntax	cedg 26346	Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).
class Edg
Definition	df-edg 26347	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 (e.g., edguhgr 26428). 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))
Theorem	edgval 26348	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	iedgedg 26349	An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.)
⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺))
Theorem	edgopval 26350	The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
Theorem	edgov 26351	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 26350. 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 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉Edg𝐸) = ran 𝐸)
Theorem	edgstruct 26352	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 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸)
Theorem	edgiedgb 26353*	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)    ⇒   ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
Theorem	edg0iedg0 26354	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)    ⇒   ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))
16.2  Undirected graphs For undirected graphs, we will have the following hierarchy/taxonomy: * Undirected Hypergraph: UHGraph * Undirected loop-free graphs: ULFGraph (not defined formally yet) * Undirected simple Hypergraph: USHGraph => USHGraph ⊆ UHGraph (ushgruhgr 26368) * Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 26401) * Undirected loop-free hypergraph: ULFHGraph (not defined formally yet) => ULFHGraph ⊆ UHGraph and ULFHGraph ⊆ ULFGraph * Undirected loop-free simple hypergraph: ULFSHGraph (not defined formally yet) => ULFSHGraph ⊆ USHGraph and ULFSHGraph ⊆ ULFHGraph * Undirected simple Pseudograph: USPGraph => USPGraph ⊆ UPGraph (uspgrupgr 26476) and USPGraph ⊆ USHGraph (uspgrushgr 26475), see also uspgrupgrushgr 26477 * Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 26402) and UMGraph ⊆ ULFHGraph (umgrislfupgr 26422) * Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 26478) and USGraph ⊆ UMGraph (usgrumgr 26479) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 26484) see also usgrumgruspgr 26480
16.2.1  Undirected hypergraphs
Syntax	cuhgr 26355	Extend class notation with undirected hypergraphs.
class UHGraph
Syntax	cushgr 26356	Extend class notation with undirected simple hypergraphs.
class USHGraph
Definition	df-uhgr 26357*	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 power set 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 ∖ {∅})}
Definition	df-ushgr 26358*	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 AV, 8-Oct-2020.)
⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
Theorem	isuhgr 26359	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 ∈ 𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
Theorem	isushgr 26360	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 ∈ 𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))
Theorem	uhgrf 26361	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 ∖ {∅}))
Theorem	ushgrf 26362	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 ∈ USHGraph → 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅}))
Theorem	uhgrss 26363	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) → (𝐸‘𝐹) ⊆ 𝑉)
Theorem	uhgreq12g 26364	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 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))
Theorem	uhgrfun 26365	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)
Theorem	uhgrn0 26366	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 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅)
Theorem	lpvtx 26367	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 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))
Theorem	ushgruhgr 26368	An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
Theorem	isuhgrop 26369	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 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
Theorem	uhgr0e 26370	The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → (iEdg‘𝐺) = ∅)    ⇒   ⊢ (𝜑 → 𝐺 ∈ UHGraph)
Theorem	uhgr0vb 26371	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 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
Theorem	uhgr0 26372	The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
⊢ ∅ ∈ UHGraph
Theorem	uhgrun 26373	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 (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ UHGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))    ⇒   ⊢ (𝜑 → 𝑈 ∈ UHGraph)
Theorem	uhgrunop 26374	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 ∪ 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.)
⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ UHGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    ⇒   ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph)
Theorem	ushgrun 26375	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 (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)
⊢ (𝜑 → 𝐺 ∈ USHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USHGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))    ⇒   ⊢ (𝜑 → 𝑈 ∈ UHGraph)
Theorem	ushgrunop 26376	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 ∪ 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.)
⊢ (𝜑 → 𝐺 ∈ USHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USHGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    ⇒   ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph)
Theorem	uhgrstrrepe 26377	Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
⊢ 𝑉 = (Base‘𝐺)    &   ⊢ 𝐼 = (.ef‘ndx)    &   ⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))    ⇒   ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph)
Theorem	incistruhgr 26378*	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 ⊆ (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 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 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph))
16.2.2  Undirected pseudographs and multigraphs
Syntax	cupgr 26379	Extend class notation with undirected pseudographs.
class UPGraph
Syntax	cumgr 26380	Extend class notation with undirected multigraphs.
class UMGraph
Definition	df-upgr 26381*	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 26382). (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}}
Definition	df-umgr 26382*	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 13545 and isumgrs 26395). (Contributed by AV, 24-Nov-2020.)
⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
Theorem	isupgr 26383*	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 ∈ 𝑈 → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
Theorem	wrdupgr 26384*	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 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐺 ∈ UPGraph ↔ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
Theorem	upgrf 26385*	The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 26386 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})
Theorem	upgrfn 26386*	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})
Theorem	upgrss 26387	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) → (𝐸‘𝐹) ⊆ 𝑉)
Theorem	upgrn0 26388	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 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅)
Theorem	upgrle 26389	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 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2)
Theorem	upgrfi 26390	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 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ Fin)
Theorem	upgrex 26391*	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 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
Theorem	upgrbi 26392*	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.)
⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ {𝑋, 𝑌} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
Theorem	upgrop 26393	A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.)
⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)
Theorem	isumgr 26394*	The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
Theorem	isumgrs 26395*	The simplified property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
Theorem	wrdumgr 26396*	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 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
Theorem	umgrf 26397*	The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn 26398 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})
Theorem	umgrfn 26398*	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})
Theorem	umgredg2 26399	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) → (♯‘(𝐸‘𝑋)) = 2)
Theorem	umgrbi 26400*	Show that an unordered pair is a valid edge in a multigraph. (Contributed by AV, 9-Mar-2021.)
⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑋 ≠ 𝑌    ⇒   ⊢ {𝑋, 𝑌} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}