P. 291 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

axcontlem5 29001*

Lemma for axcont 29009. Compute the value of 𝐹. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ 𝑃 ∈ 𝐷) → ((𝐹‘𝑃) = 𝑇 ↔ (𝑇 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑃‘𝑖) = (((1 − 𝑇) · (𝑍‘𝑖)) + (𝑇 · (𝑈‘𝑖))))))

Theorem

axcontlem6 29002*

Lemma for axcont 29009. State the defining properties of the value of 𝐹. (Contributed by Scott Fenton, 19-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ 𝑃 ∈ 𝐷) → ((𝐹‘𝑃) ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑃‘𝑖) = (((1 − (𝐹‘𝑃)) · (𝑍‘𝑖)) + ((𝐹‘𝑃) · (𝑈‘𝑖)))))

Theorem

axcontlem7 29003*

Lemma for axcont 29009. Given two points in 𝐷, one preceeds the other iff its scaling constant is less than the other point's. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ (𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷)) → (𝑃 Btwn ⟨𝑍, 𝑄⟩ ↔ (𝐹‘𝑃) ≤ (𝐹‘𝑄)))

Theorem

axcontlem8 29004*

Lemma for axcont 29009. A point in 𝐷 is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ (𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷)) → (((𝐹‘𝑃) ≤ (𝐹‘𝑄) ∧ (𝐹‘𝑄) ≤ (𝐹‘𝑅)) → 𝑄 Btwn ⟨𝑃, 𝑅⟩))

Theorem

axcontlem9 29005*

Lemma for axcont 29009. Given the separation assumption, all values of 𝐹 over 𝐴 are less than or equal to all values of 𝐹 over 𝐵. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∀𝑛 ∈ (𝐹 “ 𝐴)∀𝑚 ∈ (𝐹 “ 𝐵)𝑛 ≤ 𝑚)

Theorem

axcontlem10 29006*

Lemma for axcont 29009. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theorem

axcontlem11 29007*

Lemma for axcont 29009. Eliminate the hypotheses from axcontlem10 29006. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theorem

axcontlem12 29008*

Lemma for axcont 29009. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ 𝑍 ∈ (𝔼‘𝑁)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theorem

axcont 29009*

The axiom of continuity. Take two sets of points 𝐴 and 𝐵. If all the points in 𝐴 come before the points of 𝐵 on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

16.4.2.3  EE^n fulfills Tarski's Axioms

Syntax

ceeng 29010

Extends class notation with the Tarski geometry structure for 𝔼↑𝑁.

class EEG

Definition

df-eeng 29011*

Define the geometry structure for 𝔼↑𝑁. (Contributed by Thierry Arnoux, 24-Aug-2017.)

⊢ EEG = (𝑛 ∈ ℕ ↦ ({⟨(Base‘ndx), (𝔼‘𝑛)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ ((𝔼‘𝑛) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theorem

eengv 29012*

The value of the Euclidean geometry for dimension 𝑁. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) = ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theorem

eengstr 29013

The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, ;17⟩)

Theorem

eengbas 29014

The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))

Theorem

ebtwntg 29015

The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (Itv‘(EEG‘𝑁))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑍 Btwn ⟨𝑋, 𝑌⟩ ↔ 𝑍 ∈ (𝑋𝐼𝑌)))

Theorem

ecgrtg 29016

The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ − = (dist‘(EEG‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)))

Theorem

elntg 29017*

The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)

⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (Itv‘(EEG‘𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))

Theorem

elntg2 29018*

The line definition in the Tarski structure for the Euclidean geometry. In contrast to elntg 29017, the betweenness can be strengthened by excluding 1 resp. 0 from the related intervals (because of 𝑥 ≠ 𝑦). (Contributed by AV, 14-Feb-2023.)

⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (1...𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))}))

Theorem

eengtrkg 29019

The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Theorem

eengtrkge 29020

The geometry structure for 𝔼↑𝑁 is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_E)

PART 17  GRAPH THEORY

To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to definitions in [Bollobas] p. 1-8 or in [Diestel] p. 2-28. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic concepts:

Term	Reference	Definition	Remarks
Vertex	df-vtx 29033	A vertex of a graph 𝐺 is an element of the set of vertices (Vtx‘𝐺) of the graph 𝐺.	The set of vertices (Vtx‘𝐺) (corresponding to V(G) in [Bollobas] p. 1) is usually the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩ (see opvtxfv 29039), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 29051).
Edge	df-edg 29083	An edge of a graph 𝐺 is a nonempty set of vertices of the graph. It is said that these vertices are "joined" or "connected" by the edge, see [Bollobas] p. 1.	The set of edges (Edg‘𝐺) (corresponding to E(G) in [Bollobas] p. 1) is usually the range ran 𝐸 of the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩, or the range of the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure.
Loop		A loop in a graph 𝐺 is an edge which connects a single vertex with itself (or, according to [Bollobas] p. 7 "joins a vertex to itself").	In other words, a loop is an edge 𝑒 ∈ (Edg‘𝐺) which is a singleton consisting of a vertex 𝑣 ∈ (Vtx‘𝐺): 𝑒 = {𝑣}
Edge function resp. indexed edges	df-iedg 29034	An edge function (or indexed set of edges) of a graph 𝐺 is a mapping of an arbitrary index set to nonempty sets of vertices of the graph.	The edge function (iEdg‘𝐺) is usually the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸 > (see opiedgfv 29042), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 29052). The set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 29084. Whereas the concept of plain edges is sufficient for simple hypergraphs, indexed edges are required for e.g., multigraphs in which the same vertices may be connected by more than one edge.

Basic kinds of graphs:

Term	Reference	Definition	Remarks
Undirected hypergraph	df-uhgr 29093	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}).	In this most general definition of a graph, an "edge" may connect three or more vertices with each other, see [Berge] p. 1. In Wikipedia "Hypergraph", see https://en.wikipedia.org/wiki/Hypergraph 29093 (18-Jan-2020) such a hypergraph is called a "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure 29093 (18-Jan-2020) "Each hypergraph [...] can be regarded as an incidence structure in which the [vertices] play the role of "points", the corresponding family of [edges] plays the role of "lines" and the incidence relation is set membership". Notice that by using (Edg‘𝐺) the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this representation will only be used for undirected simple hypergraphs.
Undirected simple hypergraph	df-ushgr 29094	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}).	See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph 29094 (18-Jan-2020). This is how a "hypergraph" is defined in Section I.1 in [Bollobas] p. 7 or the definition in section 1.10 in [Diestel] p. 27. A simple hypergraph has at most one edge between the same vertices, hence a pseudograph needs not be a simple hypergraph. According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E_1, E_2, ..., E_m } such that E_i C_ E_j => i = j". By this definition, a simple hypergraph cannot contain the edges E_1 = { v_1 , v_2 } and E_2 = { v_1, v_2, v_3 }, because E_1 C_ E_2, but 1 =/= 2.
Undirected loop-free hypergraph	---	an undirected hypergraph without a loop, i.e. all edges connect at least two vertices.
Undirected pseudograph	df-upgr 29117	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 4886). This means a pseudograph may contain loops. This definition corresponds to the definition of a "multigraph" in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed", or in [Diestel] p. 28, "A multigraph is a pair (V,E) of disjoint sets (of vertices and edges) together with a map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end(s).".
Undirected multigraph	df-umgr 29118	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	This definition 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." A proper unordered pair contains two different elements, therefore a multigraph does not have loops.
Undirected simple pseudograph	df-uspgr 29185	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph	df-usgr 29186	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	An ordered pair ⟨𝑉, 𝐸⟩ of two distinct sets 𝑉 (the vertices) and 𝐸 (the edges), the "usual" definition of a "graph", see, for example, the definition in section I.1 of [Bollobas] p. 1 or in section 1.1 of [Diestel] p. 2, can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see usgrausgrb 29204.
Finite graph	df-fusgr 29352	a graph 𝐺 with a finite set of vertices 𝑉 = (Vtx‘𝐺).	See definitions in [Bollobas] p. 1 or [Diestel] p. 2. In simple graphs, the set of (indexed) edges (iEdg‘𝐺) (and therefore also the set of edges (Edg‘𝐺)) is finite if 𝑉 = (Vtx‘𝐺) is finite, see fusgrfis 29365. The number of edges is limited by (𝑛C2) (or "𝑛 choose 2") with 𝑛 = (♯‘𝑉), see fusgrmaxsize 29500. Analogously, the number of edges 𝐸 = (iEdg‘𝐺) of an undirected simple pseudograph (which may have loops) is limited by ((𝑛 + 1)C2). In pseudographs or multigraphs, however, 𝐸 can be infinite although 𝑉 is finite.
Graph of finite size	---	a graph 𝐺 with a finite set 𝐸 = (iEdg‘𝐺), i.e. with a finite number of edges.	A graph can be of finite size although its set of vertices is infinite (most of the vertices would not be connected by an edge).

Terms and properties of graphs:

Term	Reference	Definition	Remarks
Edge joining resp. connecting (two) vertices	---	An edge 𝑒 ∈ (Edg‘𝐺) joins resp. connects the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) if 𝑒 = { v_1, v_2, ... v_n }.	If 𝑛 = 1, 𝑒 = { v_1 } is a loop, if 𝑛 = 2, 𝑒 = { v_1 , v_2 } is an edge as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge	see definition in Section I.1 in [Bollobas] p. 1.	If an edge 𝑒 ∈ (Edg‘𝐺) joins the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ), then the vertices v_1, v_2, ... v_n are called the endvertices of the edge 𝑒.
(Two) Adjacent vertices	see definition in Section I.1 in [Bollobas] p. 1/2.	The vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) are adjacent if there is an edge e = { v_1, v_2, ... v_n } joining these vertices. In this case, the vertices are incident with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or connected by the edge e.
Edge ending at a vertex		An edge 𝑒 ∈ (Edg‘𝐺) is ending at a vertex 𝑣 if the vertex is an endvertex of the edge: 𝑣 ∈ 𝑒. In other words, the vertex 𝑣 is incident with the edge 𝑒.
(Two) Adjacent edges		The edges e_0, e_1, ... e_n (𝑛 ∈ ℕ) are adjacent if they have exactly one common endvertex.	Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph	see definition in Section I.1 in [Bollobas] p. 3	The order of a graph 𝐺 is the number of vertices in the graph: (♯‘(Vtx‘𝐺)).
Size of a graph	see definition in Section I.1 in [Bollobas] p. 3	The size of a graph 𝐺 is the number of edges in the graph: (♯‘(iEdg‘𝐺)). Or, for a simple graph 𝐺: (♯‘(Edg‘𝐺))).
Neighborhood of a vertex	df-nbgr 29368 resp. definition in Section I.1 in [Bollobas] p. 3	A vertex connected with a vertex 𝑣 by an edge is called a neighbor of the vertex 𝑣. The set of neighbors of a vertex 𝑣 is called the neighborhood (or open neighborhood) of the vertex 𝑣. The closed neighborhood is the union of the (open) neighborhood of the vertex 𝑣 with {𝑣} (see df-clnbgr 47693).
Degree of a vertex	df-vtxdg 29502	The degree of a vertex is the number of the edges ending at this vertex.	In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex	usgrvd0nedg 29569	A vertex is called isolated if it is not an endvertex of any edge, thus having degree 0.
Universal vertex	df-uvtx 29421	A vertex is called universal if it is connected with every other vertex of the graph by an edge, thus having degree ((♯‘(Vtx‘𝐺)) − ).

Special kinds of graphs:

Term	Reference	Definition	Remarks
Complete graph	df-cplgr 29446	A graph is called complete if each pair of vertices is connected by an edge.	The size of a complete undirected simple graph of order 𝑛 is (𝑛C2) (or "𝑛 choose 2"), see cusgrsize 29490.
Empty graph	uhgr0e 29106	A graph is called empty if it has no edges.
Null graph	uhgr0 29108 and uhgr0vb 29107	A graph is called a null graph if it has no vertices (and therefore also no edges).
Trivial graph	usgr1v 29291	A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph	df-conngr 30219 resp. definition in Section I.1 in [Bollobas] p. 6	A graph is called connected if for each pair of vertices there is a path between these vertices.

For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.

17.1  Vertices and edges

In the following, the vertices and (indexed) edges for an arbitrary class 𝐺 (called "graph" in the following) are defined and examined. The main result of this section is to show that the set of vertices (Vtx‘𝐺) of a graph 𝐺 is the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩ (see opvtxfv 29039), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 29051), and that the set of indexed edges resp. the edge function (iEdg‘𝐺) is the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩ (see opiedgfv 29042), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 29052). Finally, it is shown that the set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 29084.

Usually, a graph 𝐺 is a set. If 𝐺 is a proper class, however, it represents the null graph (without vertices and edges), because (Vtx‘𝐺) = ∅ and (iEdg‘𝐺) = ∅ holds, see vtxvalprc 29080 and iedgvalprc 29081.

Up to the end of this section, the edges need not be related to the vertices. Once undirected hypergraphs are defined (see df-uhgr 29093), the edges become nonempty sets of vertices, and by this obtain their meaning as "connectors" of vertices.

17.1.1  The edge function extractor for extensible structures

Syntax

cedgf 29021

Extend class notation with an edge function.

class .ef

Definition

df-edgf 29022

Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.) Use its index-independent form edgfid 29023 instead. (New usage is discouraged.)

⊢ .ef = Slot ;18

Theorem

edgfid 29023

Utility theorem: index-independent form of df-edgf 29022. (Contributed by AV, 16-Nov-2021.)

⊢ .ef = Slot (.ef‘ndx)

Theorem

edgfndx 29024

Index value of the df-edgf 29022 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)

⊢ (.ef‘ndx) = ;18

Theorem

edgfndxnn 29025

The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)

⊢ (.ef‘ndx) ∈ ℕ

Theorem

edgfndxid 29026

The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 28-Oct-2024.)

⊢ (𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorem

edgfndxidOLD 29027

Obsolete version of edgfndxid 29026 as of 28-Oct-2024. The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorem

basendxltedgfndx 29028

The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 30-Oct-2024.)

⊢ (Base‘ndx) < (.ef‘ndx)

Theorem

baseltedgfOLD 29029

Obsolete proof of basendxltedgfndx 29028 as of 30-Oct-2024. The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Base‘ndx) < (.ef‘ndx)

Theorem

basendxnedgfndx 29030

The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)

⊢ (Base‘ndx) ≠ (.ef‘ndx)

17.1.2  Vertices and indexed edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/pseudo-/ multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath: ⟨𝑉, 𝐸⟩.

Another way is to regard a graph as a mathematical structure, which consists at least of a set (of vertices) and a relation between the vertices (edge function), but which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 17194.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 29033 and df-iedg 29034), which provide the vertices resp. (indexed) edges of an arbitrary class 𝐺 which represents a graph: (Vtx‘𝐺) resp. (iEdg‘𝐺). In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G).").

Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. For example, e₁ = e(1) = { a, b } and e₂ = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e₁ and e₂ connecting the same two vertices a and b, we would have e(e₁) = e(e₂) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair (𝐺 ∈ (V × V)), the set of vertices is (Vtx‘𝐺) = (1^st ‘𝐺) (see opvtxval 29038) and the set of (indexed) edges is (iEdg‘𝐺) = (2^nd ‘𝐺) (see opiedgval 29041), or if 𝐺 is given as ordered pair 𝐺 = ⟨𝑉, 𝐸⟩, the set of vertices is (Vtx‘𝐺) = 𝑉 (see opvtxfv 29039) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see opiedgfv 29042).

And for a graph represented as extensible structure (𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩), the set of vertices is (Vtx‘𝐺) = (Base‘𝐺) (see funvtxval 29053) and the set of (indexed) edges is (iEdg‘𝐺) = (.ef‘𝐺) (see funiedgval 29054), or if 𝐺 is given in its simplest form as extensible structure with two slots (𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), the set of vertices is (Vtx‘𝐺) = 𝑉 (see struct2grvtx 29062) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see struct2griedg 29063).

These two representations are convertible, see graop 29064 and grastruct 29065: If 𝐺 is a graph (for example 𝐺 = ⟨𝑉, 𝐸⟩), then 𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩} represents essentially the same graph, and if 𝐺 is a graph (for example 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), then 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ represents essentially the same graph. In both cases, (Vtx‘𝐺) = (Vtx‘𝐻) and (iEdg‘𝐺) = (iEdg‘𝐻) hold. Theorems gropd 29066 and gropeld 29068 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, 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. Analogously, theorems grstructd 29067 and grstructeld 29069 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then any extensible structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is also such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be 𝐺 = ⟨𝑉, ∅⟩ or 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), ∅⟩}, a structure without a slot for edges can be used: 𝐺 = {⟨(Base‘ndx), 𝑉⟩}, see snstrvtxval 29072 and snstriedgval 29073. Analogously, the empty set ∅ can be used to represent the null graph, see vtxval0 29074 and iedgval0 29075, which can also be represented by 𝐺 = ⟨∅, ∅⟩ or 𝐺 = {⟨(Base‘ndx), ∅⟩, ⟨(.ef‘ndx), ∅⟩}. Even proper classes can be used to represent the null graph, see vtxvalprc 29080 and iedgvalprc 29081.

Other classes should not be used to represent graphs, because there could be a degenerate behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 29076 resp. iedgvalsnop 29077, and vtxval3sn 29078 resp. iedgval3sn 29079. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction of ordered pairs, see also the comment for df-op 4655.

17.1.2.1  Definitions and basic properties

Syntax

cvtx 29031

Extend class notation with the vertices of "graphs".

class Vtx

Syntax

ciedg 29032

Extend class notation with the indexed edges of "graphs".

class iEdg

Definition

df-vtx 29033

Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)

⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))

Definition

df-iedg 29034

Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)

⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^nd ‘𝑔), (.ef‘𝑔)))

Theorem

vtxval 29035

The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺))

Theorem

iedgval 29036

The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)

⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2^nd ‘𝐺), (.ef‘𝐺))

Theorem

1vgrex 29037

A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

17.1.2.2  The vertices and edges of a graph represented as ordered pair

Theorem

opvtxval 29038

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) = (1^st ‘𝐺))

Theorem

opvtxfv 29039

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 29040

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 29041

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) = (2^nd ‘𝐺))

Theorem

opiedgfv 29042

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 29043

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 29044

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 29045

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

17.1.2.3  The vertices and edges of a graph represented as extensible structure

Theorem

funvtxdmge2val 29046

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 29047

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 29048

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 29049

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 29050

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 29051

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 29052

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 29053

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 29054

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 29055

Lemma for structvtxval 29056 and structiedg0val 29057. (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 29056

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 29057

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 29058

Lemma for structgrssvtx 29059 and structgrssiedg 29060. (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 29059

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 29060

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 29061

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 29062

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 29063

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 29064

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 29065

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 29066*

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 29067*

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 29068*

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 29069*

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 29070

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 29071

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)

17.1.2.4  Representations of graphs without edges

Theorem

snstrvtxval 29072

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 29076 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 29073

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 29077 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) = ∅)

17.1.2.5  Degenerated cases of representations of graphs

Theorem

vtxval0 29074

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 29075

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 29076

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 29077

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 29078

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 29079

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 29080

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 29081

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) = ∅)

17.1.3  Edges as range of the edge function

Syntax

cedg 29082

Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).

class Edg

Definition

df-edg 29083

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 29164). 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 29084

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 29085

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 29086

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 29087

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 29086. 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 29088

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 29089*

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 29090

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 = ∅))

17.2  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 29104)

* Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 29137)

* 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 29213) and USPGraph ⊆ USHGraph (uspgrushgr 29212), see also uspgrupgrushgr 29214

* Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 29138) and UMGraph ⊆ ULFHGraph (umgrislfupgr 29158)

* Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 29215) and USGraph ⊆ UMGraph (usgrumgr 29216) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 29222) see also usgrumgruspgr 29217

17.2.1  Undirected hypergraphs

Syntax

cuhgr 29091

Extend class notation with undirected hypergraphs.

class UHGraph

Syntax

cushgr 29092

Extend class notation with undirected simple hypergraphs.

class USHGraph

Definition

df-uhgr 29093*

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 29094*

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 29095

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 29096

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 29097

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 29098

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 29099

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 29100

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))