P. 288 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cffldtocusgr 28701*	The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2}    &   ⊢ 𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)    ⇒   ⊢ 𝐺 ∈ ComplUSGraph
Theorem	cusgrres 28702*	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 28703	Base case of the induction in cusgrsize 28708. 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 28704	Base case of the induction in cusgrsize 28708. 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 28705*	Lemma for cusgrsizeinds 28706. (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 28706*	Part 1 of induction step in cusgrsize 28708. 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 28707*	Induction step in cusgrsize 28708. 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 28708	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 28709*	Lemma 1 for cusgrfi 28712. (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 28710*	Lemma 2 for cusgrfi 28712. (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 28711*	Lemma 3 for cusgrfi 28712. (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 28712	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 28713	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 28714*	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 28715	Lemma 1 for sizusglecusg 28717. (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 28716	Lemma 2 for sizusglecusg 28717. (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 28717	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 28718	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 28719	Extend class notation with the vertex degree function.
class VtxDeg
Definition	df-vtxdg 28720*	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 28721*	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 28722*	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 28723*	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 28724	The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
Theorem	vtxdgf 28725	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 28726	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 28727	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 28728	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 28790, vdegp1bi 28791 and vdegp1ci 28792). (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 28729	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 28730	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 28731	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 28732	The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 28728. (Contributed by AV, 15-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉 ∧ 𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Theorem	vtxdlfuhgr1v 28733*	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 28734	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 28735	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 28736	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 28737	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 28738	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 28739*	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 28740*	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 28741*	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 28742*	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 28743*	Lemma for vtxdushgrfvedg 28744 and vtxdusgrfvedg 28745. (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 28744*	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 28745*	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 28746*	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 28747*	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 28748*	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 28749*	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 28750*	Alternate proof of vtxdusgr0edgnel 28749, not based on vtxduhgr0edgnel 28748. 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 28751	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 28752*	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 28753*	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 28754	A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 28503). (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 28755	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 28756	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 28757	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 28758	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 28759	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 28760	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 28761	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 28762	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 28763	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 28764	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 28765	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 28766	Lemma for p1evtxdeq 28767 and p1evtxdp1 28768. (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 28767	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 28768	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 28769	The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 28503). (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
Theorem	uspgrloopvtxel 28770	A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 28503). (Contributed by AV, 17-Dec-2020.)
⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘𝐺))
Theorem	uspgrloopiedg 28771	The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 28503) 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 28772	The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 28503) 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 28773	In a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 28503), 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 28774	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 28503), 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 28775	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 28776	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 28777	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 28778	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 28779	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 28780	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 28781	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 28782	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 28773, but degree 2, see uspgrloopvd2 28774. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 28780, but also degree 2, see umgr2v2evd2 28781. (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 28783	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 28784*	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 28785*	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 28786*	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 28787*	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 28788*	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 28789*	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 28790*	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 28791*	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 28792*	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 28793	Lemma 1 for vtxdginducedm1 28797: 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 28794*	Lemma 2 for vtxdginducedm1 28797: 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 28795*	Lemma 3 for vtxdginducedm1 28797: 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 28796*	Lemma 4 for vtxdginducedm1 28797. (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 28797*	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 28798*	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 28799*	Lemma for finsumvtxdg2sstep 28803. (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 28800*	Lemma for finsumvtxdg2sstep 28803. (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}})))