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Theorem List for Metamath Proof Explorer - 40601-40700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremissubgr2 40601 The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝑆)       ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))
 
Theoremsubgrprop 40602 The properties of a subgraph. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝑆)       (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
 
Theoremsubgrprop2 40603 The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺, connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝑆)       (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉))
 
Theoremuhgrissubgr 40604 The property of a hypergraph to be a subgraph. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)       ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))
 
Theoremsubgrprop3 40605 The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝑆)    &   𝐵 = (Edg‘𝐺)       (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))
 
Theoremegrsubgr 40606 An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 17-Dec-2020.)
(((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
 
Theorem0grsubgr 40607 The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
(𝐺𝑊 → ∅ SubGraph 𝐺)
 
Theorem0uhgrsubgr 40608 The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.)
((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)
 
Theoremuhgrsubgrself 40609 A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
(𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)
 
Theoremsubgrfun 40610 The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.)
((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
 
Theoremsubgruhgrfun 40611 The edge function of a subgraph of a hypergraph is a function. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 20-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
 
Theoremsubgreldmiedg 40612 An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.)
((𝑆 SubGraph 𝐺𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
 
Theoremsubgruhgredgd 40613 An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐼 = (iEdg‘𝑆)    &   (𝜑𝐺 ∈ UHGraph )    &   (𝜑𝑆 SubGraph 𝐺)    &   (𝜑𝑋 ∈ dom 𝐼)       (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
 
Theoremsubumgredg2 40614* An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐼 = (iEdg‘𝑆)       ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
 
Theoremsubuhgr 40615 A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph )
 
Theoremsubupgr 40616 A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph )
 
Theoremsubumgr 40617 A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.)
((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph )
 
Theoremsubusgr 40618 A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.)
((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph )
 
Theoremuhgrspansubgrlem 40619 Lemma for uhgrspansubgr 40620: The edges of the graph 𝑆 obtained by removing some edges of a hypergraph 𝐺 are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 40620. (Contributed by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph )       (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
 
Theoremuhgrspansubgr 40620 A spanning subgraph 𝑆 of a hypergraph 𝐺 is actually a subgraph of 𝐺. A subgraph 𝑆 of a graph 𝐺 which has the same vertices as 𝐺 and is obtained by removing some edges of 𝐺 is called a spanning subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). Formally, the edges are "removed" by restricting the edge function of the original graph by an arbitrary class (which actually needs not to be a subset of the domain of the edge function). (Contributed by AV, 18-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph )       (𝜑𝑆 SubGraph 𝐺)
 
Theoremuhgrspan 40621 A spanning subgraph 𝑆 of a hypergraph 𝐺 is a hypergraph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph )       (𝜑𝑆 ∈ UHGraph )
 
Theoremupgrspan 40622 A spanning subgraph 𝑆 of a pseudograph 𝐺 is a pseudograph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UPGraph )       (𝜑𝑆 ∈ UPGraph )
 
Theoremumgrspan 40623 A spanning subgraph 𝑆 of a multigraph 𝐺 is a multigraph. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UMGraph )       (𝜑𝑆 ∈ UMGraph )
 
Theoremusgrspan 40624 A spanning subgraph 𝑆 of a simple graph 𝐺 is a simple graph. (Contributed by AV, 15-Oct-2020.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ USGraph )       (𝜑𝑆 ∈ USGraph )
 
Theoremuhgrspanop 40625 A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph )
 
Theoremupgrspanop 40626 A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UPGraph )
 
Theoremumgrspanop 40627 A spanning subgraph of a multigraph represented by an ordered pair is a multigraph. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UMGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UMGraph )
 
Theoremusgrspanop 40628 A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ USGraph )
 
Theoremuhgrspan1lem1 40629 Lemma 1 for uhgrspan1 40632. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}       ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
 
Theoremuhgrspan1lem2 40630 Lemma 2 for uhgrspan1 40632. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
 
Theoremuhgrspan1lem3 40631 Lemma 3 for uhgrspan1 40632. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       (iEdg‘𝑆) = (𝐼𝐹)
 
Theoremuhgrspan1 40632* The induced subgraph 𝑆 of a hypergraph 𝐺 obtained by removing one vertex is actually a subgraph of 𝐺. A subgraph is called induced or spanned by a subset of vertices of a graph if it contains all edges of the original graph that join two vertices of the subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 𝑆 SubGraph 𝐺)
 
Theoremupgrres1lem1 40633* Lemma 1 for upgrres1 40637. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
 
Theoremumgrres1lem 40634* Lemma for umgrres1 40638. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
 
Theoremupgrres1lem2 40635* Lemma 2 for upgrres1 40637. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
 
Theoremupgrres1lem3 40636* Lemma 3 for upgrres1 40637. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (iEdg‘𝑆) = ( I ↾ 𝐹)
 
Theoremupgrres1 40637* A pseudograph obtained by removing one vertex and all edges incident with this vertex is a pseudograph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 40597 since the domains of the edge functions may not be compatible. (Contributed by AV, 8-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph )
 
Theoremumgrres1 40638* A multigraph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 40597 since the domains of the edge functions may not be compatible. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝑆 ∈ UMGraph )
 
Theoremusgrres1 40639* Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted graph is not a subgraph of the original graph in the sense of df-subgr 40597 since the domains of the edge functions may not be compatible. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph )
 
20.34.8.10  Undirected simple graphs - finite graphs
 
Syntaxcfusgr 40640 Extend class notation with finite simple graphs.
class FinUSGraph
 
Definitiondf-fusgr 40641 Define the class of all finite undirected simple graphs without loops (called "finite simple graphs" in the following). A finite graph is an undirected simple graph of finite order, i.e. with a finite set of vertices. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
 
Theoremisfusgr 40642 The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
 
Theoremfusgrvtxfi 40643 A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
 
Theoremisfusgrf1 40644* The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (𝐺 ∈ FinUSGraph ↔ (𝐼:dom 𝐼1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ 𝑉 ∈ Fin)))
 
Theoremisfusgrcl 40645 The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 9-Jan-2020.)
(𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (#‘(Vtx‘𝐺)) ∈ ℕ0))
 
Theoremfusgrusgr 40646 A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.)
(𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
 
Theoremopfusgr 40647 A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.)
((𝑉𝑋𝐸𝑌) → (⟨𝑉, 𝐸⟩ ∈ FinUSGraph ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))
 
Theoremusgredgffibi 40648 The number of edges in a simple graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 22-Oct-2020.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (𝐸 ∈ Fin ↔ 𝐼 ∈ Fin))
 
Theoremfusgredgfi 40649* In a finite simple graph the number of edges which contain a given vertex is also finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 21-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ∈ Fin)
 
Theoremusgr1v0e 40650 The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ (#‘𝑉) = 1) → (#‘𝐸) = 0)
 
Theoremusgrfilem 40651* In a finite simple graph, the number of edges is finite iff the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin))
 
Theoremfusgrfisbase 40652 Induction base for fusgrfis 40654. Main work is done in uhgr0v0e 40569. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.)
(((𝑉𝑋𝐸𝑌) ∧ ⟨𝑉, 𝐸⟩ ∈ USGraph ∧ (#‘𝑉) = 0) → 𝐸 ∈ Fin)
 
Theoremfusgrfisstep 40653* Induction step in fusgrfis 40654: In a finite simple graph, the number of edges is finite if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.)
(((𝑉𝑋𝐸𝑌) ∧ ⟨𝑉, 𝐸⟩ ∈ FinUSGraph ∧ 𝑁𝑉) → (( I ↾ {𝑝 ∈ (Edg‘⟨𝑉, 𝐸⟩) ∣ 𝑁𝑝}) ∈ Fin → 𝐸 ∈ Fin))
 
Theoremfusgrfis 40654 A finite simple graph is of finite size, i.e. has a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 8-Nov-2020.)
(𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
 
20.34.8.11  Neighbors, complete graphs and universal vertices
 
Syntaxcnbgr 40655 Extend class notation with neighbors (of a vertex in a graph).
class NeighbVtx
 
Syntaxcuvtxa 40656 Extend class notation with the universal vertices (in a graph).
class UnivVtx
 
Syntaxccplgr 40657 Extend class notation with (arbitrary) complete graphs.
class ComplGraph
 
Syntaxccusgr 40658 Extend class notation with complete simple graphs.
class ComplUSGraph
 
Definitiondf-nbgr 40659* Define the (open) neighborhood resp. the class of all neighbors of a vertex (in a graph), see definition in section I.1 of [Bollobas] p. 3 or definition in section 1.1 of [Diestel] p. 3. The neighborhood/neighbors of a vertex are all (other) vertices which are connected with this vertex by an edge. In contrast to a closed neighborhood, a vertex is not a neighbor of itself. This definition is applicable even for arbitrary hypergraphs.

Remark: To distinguish this definition from other definitions for neighborhoods resp. neighbors (e.g., nei in Topology, see df-nei 20615), the suffix Vtx is added to the class constant NeighbVtx. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.)

NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
 
Theoremnbgrprc0 40660 The set of neighbors is empty if the graph or the vertex are proper classes. (Contributed by AV, 26-Oct-2020.)
(¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
 
Definitiondf-uvtxa 40661* Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph) resp. all other vertices are its neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.)
UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
 
Definitiondf-cplgr 40662* Define the class of all complete "graphs". A class/graph is called complete if every pair of distinct vertices is connected by an edge, i.e. each vertex has all other vertices as neighbors. (Contributed by AV, 24-Oct-2020.)
ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)}
 
Definitiondf-cusgr 40663 Define the class of all complete simple graphs. A simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge, see definition in section 1.1 of [Diestel] p. 3. In contrast, the definition in section I.1 of [Bollobas] p. 3 is based on the size of (finite) complete graphs, see cusgrsize 40775. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.)
ComplUSGraph = {𝑔 ∈ USGraph ∣ 𝑔 ∈ ComplGraph}
 
Theoremnbgrcl 40664 If a class has at least one neighbor, it must be a vertex. (Contributed by AV, 6-Jun-2021.)
(𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))
 
Theoremnbgrval 40665* The set of neighbors of a vertex 𝑉 in a graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
 
Theoremdfnbgr2 40666* Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
 
Theoremdfnbgr3 40667* Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves [see also nbgraop1 25692]. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
 
Theoremnbgrnvtx0 40668 There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
 
Theoremnbgrel 40669* Characterization of a neighbor of a vertex 𝑉 in a graph 𝐺. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 6-Jun-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (𝐾 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝐾𝑉𝑁𝑉) ∧ 𝐾𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝐾} ⊆ 𝑒)))
 
Theoremnbuhgr 40670* The set of neighbors of a vertex in a hypergraph. This version of nbgrval 40665 (with 𝑁 being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
 
Theoremnbupgr 40671* The set of neighbors of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})
 
Theoremnbupgrel 40672 A neighbor of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UPGraph ∧ 𝐾𝑉) ∧ (𝑁𝑉𝑁𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸))
 
Theoremnbumgrvtx 40673* The set of neighbors of a vertex in a multigraph. (Contributed by AV, 27-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
 
Theoremnbumgr 40674* The set of neighbors of an arbitrary class in a multigraph. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ UMGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
 
Theoremnbusgrvtx 40675* The set of neighbors of a vertex in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
 
Theoremnbusgr 40676* The set of neighbors of an arbitrary class in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
 
Theoremnbgr2vtx1edg 40677* If a graph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Revised by AV, 25-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((#‘𝑉) = 2 ∧ 𝑉𝐸) → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))
 
Theoremnbuhgr2vtx1edgblem 40678* Lemma for nbuhgr2vtx1edgb 40679. This reverse direction of nbgr2vtx1edg 40677 only holds for classes whose edges are subsets of the set of vertices (hypergraphs!) (Contributed by AV, 2-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
 
Theoremnbuhgr2vtx1edgb 40679* If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 2) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
 
Theoremnbusgreledg 40680 A class/vertex is a neighbor of another class/vertex in a simple graph iff the vertices are endpoints of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.) (Revised by AV, 26-Oct-2020.)
𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸))
 
Theoremuhgrnbgr0nb 40681* A vertex which is not endpoint of an edge has no neighbor in a hypergraph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)
 
Theoremnbgr0vtxlem 40682* Lemma for nbgr0vtx 40683 and nbgr0edg 40684. (Contributed by AV, 15-Nov-2020.)
(𝜑 → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)       (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅)
 
Theoremnbgr0vtx 40683 In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
 
Theoremnbgr0edg 40684 In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
 
Theoremnbgr1vtx 40685 In a graph with one vertex, all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
((#‘(Vtx‘𝐺)) = 1 → (𝐺 NeighbVtx 𝐾) = ∅)
 
Theoremnbgrisvtx 40686 Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)
 
Theoremnbgrssvtx 40687 The neighbors of a vertex in a graph are a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑉)
 
Theoremnbgrnself 40688* A vertex in a graph is not a neighbor of itself. (Contributed by by AV, 3-Nov-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
 
Theoremusgrnbnself 40689* A vertex in a simple graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ USGraph → ∀𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣))
 
Theoremnbgrnself2 40690 A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
(𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))
 
Theoremnbgrssovtx 40691 The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 40687. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
 
Theoremnbgrssvwo2 40692 The neighbors of a vertex are a subset of all vertices except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))
 
Theoremusgrnbnself2 40693 In a simple graph, a class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.)
(𝐺 ∈ USGraph → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
 
Theoremusgrnbssovtx 40694 The neighbors of a vertex in a simple graph are a subset of all vertices of the graph except the vertex itself. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
 
Theoremusgrnbssvwo2 40695 The neighbors of a vertex in a simple graph are a subset of all vertices of the graph except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 27-Oct-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))
 
Theoremnbgrsym 40696 A vertex in a graph is a neighbor of a second vertex iff the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.)
(𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁)))
 
Theoremnbupgrres 40697* The neighborhood of a vertex in a restricted pseudograph (not necessarily valid for a hypergraph, because 𝑁, 𝐾 and 𝑀 could be connected by one edge, so 𝑀 is a neighbor of 𝐾 in the original graph, but not in the restricted graph, because the edge between 𝑀 and 𝐾, also incident with 𝑁, was removed). (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾)))
 
Theoremusgrnbcnvfv 40698 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair in a simple graph. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 27-Oct-2020.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → (𝐼‘(𝐼‘{𝐾, 𝑁})) = {𝐾, 𝑁})
 
Theoremnbusgredgeu 40699* For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝑁)) → ∃!𝑒𝐸 𝑒 = {𝑀, 𝑁})
 
Theoremedgnbusgreu 40700* For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑀)       (((𝐺 ∈ USGraph ∧ 𝑀𝑉) ∧ (𝐶𝐸𝑀𝐶)) → ∃!𝑛𝑁 𝐶 = {𝑀, 𝑛})
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