P. 294 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29301-29400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cusgrres 29301*	Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}    &   ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph)
Theorem	cusgrsizeindb0 29302	Base case of the induction in cusgrsize 29307. The size of a complete simple graph with 0 vertices, actually of every null graph, is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = ((♯‘𝑉)C2))
Theorem	cusgrsizeindb1 29303	Base case of the induction in cusgrsize 29307. The size of a (complete) simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = ((♯‘𝑉)C2))
Theorem	cusgrsizeindslem 29304*	Lemma for cusgrsizeinds 29305. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1))
Theorem	cusgrsizeinds 29305*	Part 1 of induction step in cusgrsize 29307. The size of a complete simple graph with 𝑛 vertices is (𝑛 − 1) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘𝐸) = (((♯‘𝑉) − 1) + (♯‘𝐹)))
Theorem	cusgrsize2inds 29306*	Induction step in cusgrsize 29307. If the size of the complete graph with 𝑛 vertices reduced by one vertex is "(𝑛 − 1) choose 2", the size of the complete graph with 𝑛 vertices is "𝑛 choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}    ⇒   ⊢ (𝑌 ∈ ℕ0 → ((𝐺 ∈ ComplUSGraph ∧ (♯‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉) → ((♯‘𝐹) = ((♯‘(𝑉 ∖ {𝑁}))C2) → (♯‘𝐸) = ((♯‘𝑉)C2))))
Theorem	cusgrsize 29307	The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = ((♯‘𝑉)C2))
Theorem	cusgrfilem1 29308*	Lemma 1 for cusgrfi 29311. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺))
Theorem	cusgrfilem2 29309*	Lemma 2 for cusgrfi 29311. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}    &   ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)
Theorem	cusgrfilem3 29310*	Lemma 3 for cusgrfi 29311. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}    &   ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑉 ∈ Fin ↔ 𝑃 ∈ Fin))
Theorem	cusgrfi 29311	If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin)
Theorem	usgredgsscusgredg 29312	A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → 𝐸 ⊆ 𝐹)
Theorem	usgrsscusgr 29313*	A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ∀𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 𝑒 = 𝑓)
Theorem	sizusglecusglem1 29314	Lemma 1 for sizusglecusg 29316. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸):𝐸–1-1→𝐹)
Theorem	sizusglecusglem2 29315	Lemma 2 for sizusglecusg 29316. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → 𝐸 ∈ Fin)
Theorem	sizusglecusg 29316	The size of a simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹))
Theorem	fusgrmaxsize 29317	The maximum size of a finite simple graph with 𝑛 vertices is (((𝑛 − 1)∗𝑛) / 2). See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
17.2.10  Vertex degree
Syntax	cvtxdg 29318	Extend class notation with the vertex degree function.
class VtxDeg
Definition	df-vtxdg 29319*	Define the vertex degree function for a graph. To be appropriate for arbitrary hypergraphs, we have to double-count those edges that contain 𝑢 "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is not of finite size), the extended addition +𝑒 is used for the summation of the number of "ordinary" edges" and the number of "loops". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
⊢ VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
Theorem	vtxdgfval 29320*	The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))
Theorem	vtxdgval 29321*	The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
Theorem	vtxdgfival 29322*	The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 8-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
Theorem	vtxdgop 29323	The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
Theorem	vtxdgf 29324	The vertex degree function is a function from vertices to extended nonnegative integers. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*)
Theorem	vtxdgelxnn0 29325	The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑋) ∈ ℕ0*)
Theorem	vtxdg0v 29326	The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Theorem	vtxdg0e 29327	The degree of a vertex in an empty graph is zero, because there are no edges. This is the base case for the induction for calculating the degree of a vertex, for example in a Königsberg graph (see also the induction steps vdegp1ai 29389, vdegp1bi 29390 and vdegp1ci 29391). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Theorem	vtxdgfisnn0 29328	The degree of a vertex in a graph of finite size is a nonnegative integer. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0)
Theorem	vtxdgfisf 29329	The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ0)
Theorem	vtxdeqd 29330	Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.)
⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑌)    &   ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))    &   ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))    ⇒   ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))
Theorem	vtxduhgr0e 29331	The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 29327. (Contributed by AV, 15-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉 ∧ 𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Theorem	vtxdlfuhgr1v 29332*	The degree of the vertex in a loop-free hypergraph with one vertex is 0. (Contributed by AV, 2-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0))
Theorem	vdumgr0 29333	A vertex in a multigraph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 2-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉 ∧ (♯‘𝑉) = 1) → ((VtxDeg‘𝐺)‘𝑁) = 0)
Theorem	vtxdun 29334	The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → Fun 𝐽)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxdfiun 29335	The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → Fun 𝐽)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    &   ⊢ (𝜑 → dom 𝐼 ∈ Fin)    &   ⊢ (𝜑 → dom 𝐽 ∈ Fin)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxduhgrun 29336	The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ UHGraph)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxduhgrfiun 29337	The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 7-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ UHGraph)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    &   ⊢ (𝜑 → dom 𝐼 ∈ Fin)    &   ⊢ (𝜑 → dom 𝐽 ∈ Fin)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxdlfgrval 29338*	The value of the vertex degree function for a loop-free graph 𝐺. (Contributed by AV, 23-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
Theorem	vtxdumgrval 29339*	The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 23-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
Theorem	vtxdusgrval 29340*	The value of the vertex degree function for a simple graph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
Theorem	vtxd0nedgb 29341*	A vertex has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
Theorem	vtxdushgrfvedglem 29342*	Lemma for vtxdushgrfvedg 29343 and vtxdusgrfvedg 29344. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉) → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
Theorem	vtxdushgrfvedg 29343*	The value of the vertex degree function for a simple hypergraph. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) +𝑒 (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})))
Theorem	vtxdusgrfvedg 29344*	The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
Theorem	vtxduhgr0nedg 29345*	If a vertex in a hypergraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉 ∧ (𝐷‘𝑈) = 0) → ¬ ∃𝑣 ∈ 𝑉 {𝑈, 𝑣} ∈ 𝐸)
Theorem	vtxdumgr0nedg 29346*	If a vertex in a multigraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 15-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉 ∧ (𝐷‘𝑈) = 0) → ¬ ∃𝑣 ∈ 𝑉 {𝑈, 𝑣} ∈ 𝐸)
Theorem	vtxduhgr0edgnel 29347*	A vertex in a hypergraph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 24-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))
Theorem	vtxdusgr0edgnel 29348*	A vertex in a simple graph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))
Theorem	vtxdusgr0edgnelALT 29349*	Alternate proof of vtxdusgr0edgnel 29348, not based on vtxduhgr0edgnel 29347. A vertex in a simple graph has degree 0 if there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))
Theorem	vtxdgfusgrf 29350	The vertex degree function on finite simple graphs is a function from vertices to nonnegative integers. (Contributed by AV, 12-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → (VtxDeg‘𝐺):𝑉⟶ℕ0)
Theorem	vtxdgfusgr 29351*	In a finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 12-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
Theorem	fusgrn0degnn0 29352*	In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 1-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)
Theorem	1loopgruspgr 29353	A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 29101). (Contributed by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → 𝐺 ∈ USPGraph)
Theorem	1loopgredg 29354	The set of edges in a graph (simple pseudograph) with one edge which is a loop is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})
Theorem	1loopgrnb0 29355	In a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅)
Theorem	1loopgrvd2 29356	The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = 2)
Theorem	1loopgrvd0 29357	The vertex degree of a one-edge graph, case 1 (for a loop): a loop at a vertex other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    &   ⊢ (𝜑 → 𝐾 ∈ (𝑉 ∖ {𝑁}))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0)
Theorem	1hevtxdg0 29358	The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 is 0 if 𝐷 is not incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.)
⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐷 ∉ 𝐸)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0)
Theorem	1hevtxdg1 29359	The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 (not being a loop) is 1 if 𝐷 is incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐸)    &   ⊢ (𝜑 → 2 ≤ (♯‘𝐸))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 1)
Theorem	1hegrvtxdg1 29360	The vertex degree of a graph with one hyperedge, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐵) = 1)
Theorem	1hegrvtxdg1r 29361	The vertex degree of a graph with one hyperedge, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1)
Theorem	1egrvtxdg1 29362	The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐵) = 1)
Theorem	1egrvtxdg1r 29363	The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1)
Theorem	1egrvtxdg0 29364	The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐷)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐷}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 0)
Theorem	p1evtxdeqlem 29365	Lemma for p1evtxdeq 29366 and p1evtxdp1 29367. (Contributed by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐾 ∉ dom 𝐼)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑌)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)‘𝑈)))
Theorem	p1evtxdeq 29366	If an edge 𝐸 which does not contain vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is the same in both graphs. (Contributed by AV, 2-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐾 ∉ dom 𝐼)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑈 ∉ 𝐸)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))
Theorem	p1evtxdp1 29367	If an edge 𝐸 (not being a loop) which contains vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is increased by 1. (Contributed by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐾 ∉ dom 𝐼)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐸)    &   ⊢ (𝜑 → 2 ≤ (♯‘𝐸))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1))
Theorem	uspgrloopvtx 29368	The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29101). (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
Theorem	uspgrloopvtxel 29369	A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29101). (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘𝐺))
Theorem	uspgrloopiedg 29370	The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29101) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
Theorem	uspgrloopedg 29371	The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29101) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}})
Theorem	uspgrloopnb0 29372	In a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29101), the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = ∅)
Theorem	uspgrloopvd2 29373	The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29101), the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑁) = 2)
Theorem	umgr2v2evtx 29374	The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩    ⇒   ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
Theorem	umgr2v2evtxel 29375	A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
Theorem	umgr2v2eiedg 29376	The edge function in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
Theorem	umgr2v2eedg 29377	The set of edges in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})
Theorem	umgr2v2e 29378	A multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩    ⇒   ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ UMGraph)
Theorem	umgr2v2enb1 29379	In a multigraph with two edges connecting the same two vertices, each of the vertices has one neighbor. (Contributed by AV, 18-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩    ⇒   ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐺 NeighbVtx 𝐴) = {𝐵})
Theorem	umgr2v2evd2 29380	In a multigraph with two edges connecting the same two vertices, each of the vertices has degree 2. (Contributed by AV, 18-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩    ⇒   ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ((VtxDeg‘𝐺)‘𝐴) = 2)
Theorem	hashnbusgrvd 29381	In a simple graph, the number of neighbors of a vertex is the degree of this vertex. This theorem does not hold for (simple) pseudographs, because a vertex connected with itself only by a loop has no neighbors, see uspgrloopnb0 29372, but degree 2, see uspgrloopvd2 29373. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 29379, but also degree 2, see umgr2v2evd2 29380. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = ((VtxDeg‘𝐺)‘𝑈))
Theorem	usgruvtxvdb 29382	In a finite simple graph with n vertices a vertex is universal iff the vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (𝑈 ∈ (UnivVtx‘𝐺) ↔ ((VtxDeg‘𝐺)‘𝑈) = ((♯‘𝑉) − 1)))
Theorem	vdiscusgrb 29383*	A finite simple graph with n vertices is complete iff every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 22-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → (𝐺 ∈ ComplUSGraph ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1)))
Theorem	vdiscusgr 29384*	In a finite complete simple graph with n vertices every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1) → 𝐺 ∈ ComplUSGraph))
Theorem	vtxdusgradjvtx 29385*	The degree of a vertex in a simple graph is the number of vertices adjacent to this vertex. (Contributed by Alexander van der Vekens, 9-Jul-2018.) (Revised by AV, 23-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = (♯‘{𝑣 ∈ 𝑉 ∣ {𝑈, 𝑣} ∈ 𝐸}))
Theorem	usgrvd0nedg 29386*	If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 16-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑈) = 0 → ¬ ∃𝑣 ∈ 𝑉 {𝑈, 𝑣} ∈ 𝐸))
Theorem	uhgrvd00 29387*	If every vertex in a hypergraph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UHGraph → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅))
Theorem	usgrvd00 29388*	If every vertex in a simple graph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ USGraph → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅))
Theorem	vdegp1ai 29389*	The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑌} to the edge set, where 𝑋 ≠ 𝑈 ≠ 𝑌, yields degree 𝑃 as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 ∈ 𝑉    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   ⊢ (Vtx‘𝐹) = 𝑉    &   ⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑋 ≠ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑌 ≠ 𝑈    &   ⊢ (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩)    ⇒   ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃
Theorem	vdegp1bi 29390*	The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑈, 𝑋} to the edge set, where 𝑋 ≠ 𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 ∈ 𝑉    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   ⊢ (Vtx‘𝐹) = 𝑉    &   ⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑋 ≠ 𝑈    &   ⊢ (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑈, 𝑋}”⟩)    ⇒   ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)
Theorem	vdegp1ci 29391*	The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑈} to the edge set, where 𝑋 ≠ 𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 ∈ 𝑉    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   ⊢ (Vtx‘𝐹) = 𝑉    &   ⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑋 ≠ 𝑈    &   ⊢ (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑈}”⟩)    ⇒   ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)
Theorem	vtxdginducedm1lem1 29392	Lemma 1 for vtxdginducedm1 29396: the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    ⇒   ⊢ (iEdg‘𝑆) = 𝑃
Theorem	vtxdginducedm1lem2 29393*	Lemma 2 for vtxdginducedm1 29396: the domain of the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    ⇒   ⊢ dom (iEdg‘𝑆) = 𝐼
Theorem	vtxdginducedm1lem3 29394*	Lemma 3 for vtxdginducedm1 29396: an edge in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    ⇒   ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻))
Theorem	vtxdginducedm1lem4 29395*	Lemma 4 for vtxdginducedm1 29396. (Contributed by AV, 17-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}}) = 0)
Theorem	vtxdginducedm1 29396*	The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 17-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))
Theorem	vtxdginducedm1fi 29397*	The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 of finite size obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 18-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
Theorem	finsumvtxdg2ssteplem1 29398*	Lemma for finsumvtxdg2sstep 29402. (Contributed by AV, 15-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘𝐽)))
Theorem	finsumvtxdg2ssteplem2 29399*	Lemma for finsumvtxdg2sstep 29402. (Contributed by AV, 12-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))
Theorem	finsumvtxdg2ssteplem3 29400*	Lemma for finsumvtxdg2sstep 29402. (Contributed by AV, 19-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = (♯‘𝐽))