P. 480 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47901-48000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gricbri 47901*	Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.) (Proof shortened by AV, 12-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐴)    &   ⊢ 𝑊 = (Vtx‘𝐵)    &   ⊢ 𝐼 = (iEdg‘𝐴)    &   ⊢ 𝐽 = (iEdg‘𝐵)    ⇒   ⊢ (𝐴 ≃𝑔𝑟 𝐵 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
Theorem	gricushgr 47902*	The "is isomorphic to" relation for two simple hypergraphs. (Contributed by AV, 28-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐴)    &   ⊢ 𝑊 = (Vtx‘𝐵)    &   ⊢ 𝐸 = (Edg‘𝐴)    &   ⊢ 𝐾 = (Edg‘𝐵)    ⇒   ⊢ ((𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
Theorem	gricuspgr 47903*	The "is isomorphic to" relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.) (Proof shortened by AV, 5-May-2025.)
⊢ 𝑉 = (Vtx‘𝐴)    &   ⊢ 𝑊 = (Vtx‘𝐵)    &   ⊢ 𝐸 = (Edg‘𝐴)    &   ⊢ 𝐾 = (Edg‘𝐵)    ⇒   ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))
Theorem	gricrel 47904	The "is isomorphic to" relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)
⊢ Rel ≃𝑔𝑟
Theorem	gricref 47905	Graph isomorphism is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 29-Apr-2025.)
⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑔𝑟 𝐺)
Theorem	gricsym 47906	Graph isomorphism is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 3-May-2025.)
⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑔𝑟 𝑆 → 𝑆 ≃𝑔𝑟 𝐺))
Theorem	gricsymb 47907	Graph isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Proof shortened by AV, 3-May-2025.)
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑔𝑟 𝐴))
Theorem	grictr 47908	Graph isomorphism is transitive. (Contributed by AV, 5-Dec-2022.) (Revised by AV, 3-May-2025.)
⊢ ((𝑅 ≃𝑔𝑟 𝑆 ∧ 𝑆 ≃𝑔𝑟 𝑇) → 𝑅 ≃𝑔𝑟 𝑇)
Theorem	gricer 47909	Isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 3-May-2025.) (Proof shortened by AV, 11-Jul-2025.)
⊢ ( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
Theorem	gricen 47910	Isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 3-May-2025.)
⊢ 𝐵 = (Vtx‘𝑅)    &   ⊢ 𝐶 = (Vtx‘𝑆)    ⇒   ⊢ (𝑅 ≃𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)
Theorem	opstrgric 47911	A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 4-May-2025.)
⊢ 𝐺 = ⟨𝑉, 𝐸⟩    &   ⊢ 𝐻 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 ≃𝑔𝑟 𝐻)
Theorem	ushggricedg 47912	A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩    ⇒   ⊢ (𝐺 ∈ USHGraph → 𝐺 ≃𝑔𝑟 𝐻)
Theorem	cycldlenngric 47913*	Two simple pseudographs are not isomorphic if one has a cycle and the other has no cycle of the same length. (Contributed by AV, 6-Nov-2025.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ∧ ¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ¬ 𝐺 ≃𝑔𝑟 𝐻))
Theorem	isubgrgrim 47914*	Isomorphic subgraphs induced by subsets of vertices of two graphs. (Contributed by AV, 29-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
Theorem	uhgrimisgrgriclem 47915*	Lemma for uhgrimisgrgric 47916. (Contributed by AV, 31-May-2025.)
⊢ (((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) ↔ ∃𝑘 ∈ 𝐴 ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)))
Theorem	uhgrimisgrgric 47916	For isomorphic hypergraphs, the induced subgraph of a subset of vertices of one graph is isomorphic to the subgraph induced by the image of the subset. (Contributed by AV, 31-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝑁 ⊆ 𝑉) → (𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐹 “ 𝑁)))
Theorem	clnbgrisubgrgrim 47917*	Isomorphic subgraphs induced by closed neighborhoods of vertices of two graphs. (Contributed by AV, 29-May-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑌)    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
Theorem	clnbgrgrimlem 47918*	Lemma for clnbgrgrim 47919: For two isomorphic hypergraphs, if there is an edge connecting the image of a vertex of the first graph with a vertex of the second graph, the vertex of the second graph is the image of a neighbor of the vertex of the first graph. (Contributed by AV, 2-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (Edg‘𝐻)    ⇒   ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → ((𝐾 ∈ 𝐸 ∧ {(𝐹‘𝑋), 𝑌} ⊆ 𝐾) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑌))
Theorem	clnbgrgrim 47919	Graph isomorphisms between hypergraphs map closed neighborhoods onto closed neighborhoods. (Contributed by AV, 2-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (𝐻 ClNeighbVtx (𝐹‘𝑋)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑋)))
Theorem	grimedg 47920	For two isomorphic graphs, a set of vertices is an edge in one graph iff its image by a graph isomorphism is an edge of the other graph. (Contributed by AV, 7-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 ↔ ((𝐹 “ 𝐾) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉)))
Theorem	grimedgi 47921	Graph isomorphisms map edges onto the corresponding edges. (Contributed by AV, 30-Dec-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 → (𝐹 “ 𝐾) ∈ 𝐸))
21.48.15.5  Triangles in graphs Usually, a "triangle" in graph theory is a complete graph consisting of three vertices (denoted by " K3 "), see the definition in [Diestel] p. 3 or the definition in [Bollobas] p. 5. This corresponds to the definition of a "triangle graph" (which is a more precise term) in Wikipedia "Triangle graph", https://en.wikipedia.org/wiki/Triangle_graph, 27-Jul-2025: "In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. The triangle graph is also known as the cycle graph C3 and the complete graph K3." Often, however, the term "triangle" is also used to denote a corresponding subgraph of a given graph ("triangle in a graph"), see, for example, Wikipedia "Triangle-free graph", 28-Jul-2025, https://en.wikipedia.org/wiki/Triangle-free_graph: "In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges." In this subsection, a triangle (in a graph) is defined as a set of three vertices of a given graph. In this meaning, a triangle 𝑇 with (𝑇 ∈ (GrTriangles‘𝐺)) is neither a graph nor a subgraph, but it induces a triangle graph (𝐺 ISubGr 𝑇) as subgraph of the given graph 𝐺. We require that there are three (different) edges connecting the three (different) vertices of the triangle. Therefore, it is not sufficient for arbitrary hypergraphs to say "a triangle is a set of three (different) vertices connected with each other (by edges)", because there might be only one or two multiedges fulfilling this statement. We do not regard such degenerate cases as "triangle". The definition df-grtri 47923 is designed for a special purpose, namely to provide a criterion for two graphs being not isomorphic (see grimgrtri 47934). For other purposes, a more general definition might be useful, e.g., ComplSubGr = (𝑔 ∈ V, 𝑛 ∈ ℕ ↦ {𝑡 ∈ 𝒫 𝑣 ∣ ((♯‘𝑡) = 𝑛 ∧ (𝑔 ISubGr 𝑡) ∈ ComplGraph)}) for complete subgraphs of a given size (proposed by TA). With such a definition, we would have (GrTriangles‘𝐺) = (𝐺 ComplSubGr 3) (at least for simple graphs), and the definition df-grtri 47923 may become obsolete.
Syntax	cgrtri 47922	Extend class notation with triangles (in a graph).
class GrTriangles
Definition	df-grtri 47923*	Definition of a triangles in a graph. A triangle in a graph is a set of three (different) vertices completely connected with each other. Such vertices induce a closed walk of length 3, see grtriclwlk3 47930, and the vertices of a cycle of size 3 are a triangle in a graph, see cycl3grtri 47932. (Contributed by AV, 20-Jul-2025.)
⊢ GrTriangles = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(Edg‘𝑔) / 𝑒⦌{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
Theorem	grtriproplem 47924	Lemma for grtriprop 47926. (Contributed by AV, 23-Jul-2025.)
⊢ ((𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))
Theorem	grtri 47925*	The triangles in a graph. (Contributed by AV, 20-Jul-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
Theorem	grtriprop 47926*	The properties of a triangle. (Contributed by AV, 25-Jul-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
Theorem	grtrif1o 47927	Any bijection onto a triangle preserves the edges of the triangle. (Contributed by AV, 25-Jul-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝐹:(0..^3)–1-1-onto→𝑇) → ({(𝐹‘0), (𝐹‘1)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹‘2)} ∈ 𝐸 ∧ {(𝐹‘1), (𝐹‘2)} ∈ 𝐸))
Theorem	isgrtri 47928*	A triangle in a graph. (Contributed by AV, 20-Jul-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑇 ∈ (GrTriangles‘𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
Theorem	grtrissvtx 47929	A triangle is a subset of the vertices (of a graph). (Contributed by AV, 26-Jul-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝑇 ⊆ 𝑉)
Theorem	grtriclwlk3 47930	A triangle induces a closed walk of length 3 . (Contributed by AV, 26-Jul-2025.)
⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))    &   ⊢ (𝜑 → 𝑃:(0..^3)–1-1-onto→𝑇)    ⇒   ⊢ (𝜑 → 𝑃 ∈ (3 ClWWalksN 𝐺))
Theorem	cycl3grtrilem 47931	Lemma for cycl3grtri 47932. (Contributed by AV, 5-Oct-2025.)
⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
Theorem	cycl3grtri 47932	The vertices of a cycle of size 3 are a triangle in a graph. (Contributed by AV, 5-Oct-2025.)
⊢ (𝜑 → 𝐺 ∈ UPGraph)    &   ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃)    &   ⊢ (𝜑 → (♯‘𝐹) = 3)    ⇒   ⊢ (𝜑 → ran 𝑃 ∈ (GrTriangles‘𝐺))
Theorem	grtrimap 47933	Conditions for mapping triangles onto triangles. Lemma for grimgrtri 47934 and grlimgrtri 47988. (Contributed by AV, 23-Aug-2025.)
⊢ (𝐹:𝑉–1-1→𝑊 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3)) → (((𝐹‘𝑎) ∈ 𝑊 ∧ (𝐹‘𝑏) ∈ 𝑊 ∧ (𝐹‘𝑐) ∈ 𝑊) ∧ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3)))
Theorem	grimgrtri 47934	Graph isomorphisms map triangles onto triangles. (Contributed by AV, 27-Jul-2025.) (Proof shortened by AV, 24-Aug-2025.)
⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))    ⇒   ⊢ (𝜑 → (𝐹 “ 𝑇) ∈ (GrTriangles‘𝐻))
Theorem	usgrgrtrirex 47935*	Conditions for a simple graph to contain a triangle. (Contributed by AV, 7-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 NeighbVtx 𝑎)    ⇒   ⊢ (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑁 ∃𝑐 ∈ 𝑁 (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
21.48.15.6  Star graphs According to Wikipedia "Star (graph theory)", 10-Sep-2025, https://en.wikipedia.org/wiki/Star_(graph_theory): "In graph theory, the star Sk is the complete bipartite graph K(1,k), that is, it is a tree with one internal node and k leaves. Alternatively, some authors define Sk to be the tree of order k with maximum diameter 2, in which case a star of k > 2 has k - 1 leaves.".
Syntax	cstgr 47936	Extend class notation with star graphs.
class StarGr
Definition	df-stgr 47937*	Definition of star graphs according to the first definition in Wikipedia, so that (StarGr‘𝑁) has size 𝑁, and order 𝑁 + 1: (StarGr‘0) will be a single vertex (graph without edges), see stgr0 47945, (StarGr‘1) will be a single edge (graph with two vertices connected by an edge), see stgr1 47946, and (StarGr‘3) will be a 3-star or "claw" (a star with 3 edges). (Contributed by AV, 10-Sep-2025.)
⊢ StarGr = (𝑛 ∈ ℕ0 ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
Theorem	stgrfv 47938*	The star graph SN. (Contributed by AV, 10-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) = {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩})
Theorem	stgrvtx 47939	The vertices of the star graph SN. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
Theorem	stgriedg 47940*	The indexed edges of the star graph SN. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)) = ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
Theorem	stgredg 47941*	The edges of the star graph SN. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (Edg‘(StarGr‘𝑁)) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
Theorem	stgredgel 47942*	An edge of the star graph SN. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (𝐸 ∈ (Edg‘(StarGr‘𝑁)) ↔ (𝐸 ⊆ (0...𝑁) ∧ ∃𝑥 ∈ (1...𝑁)𝐸 = {0, 𝑥})))
Theorem	stgredgiun 47943*	The edges of the star graph SN as indexed union. (Contributed by AV, 29-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (Edg‘(StarGr‘𝑁)) = ∪ 𝑥 ∈ (1...𝑁){{0, 𝑥}})
Theorem	stgrusgra 47944	The star graph SN is a simple graph. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
Theorem	stgr0 47945	The star graph S0 consists of a single vertex without edges. (Contributed by AV, 11-Sep-2025.)
⊢ (StarGr‘0) = {⟨(Base‘ndx), {0}⟩, ⟨(.ef‘ndx), ∅⟩}
Theorem	stgr1 47946	The star graph S1 consists of a single simple edge. (Contributed by AV, 11-Sep-2025.)
⊢ (StarGr‘1) = {⟨(Base‘ndx), {0, 1}⟩, ⟨(.ef‘ndx), ( I ↾ {{0, 1}})⟩}
Theorem	stgrvtx0 47947	The center ("internal node") of a star graph SN. (Contributed by AV, 12-Sep-2025.)
⊢ 𝐺 = (StarGr‘𝑁)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑉)
Theorem	stgrorder 47948	The order of a star graph SN. (Contributed by AV, 12-Sep-2025.)
⊢ 𝐺 = (StarGr‘𝑁)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (♯‘𝑉) = (𝑁 + 1))
Theorem	stgrnbgr0 47949	All vertices of a star graph SN except the center are in the (open) neighborhood of the center. (Contributed by AV, 12-Sep-2025.)
⊢ 𝐺 = (StarGr‘𝑁)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐺 NeighbVtx 0) = (𝑉 ∖ {0}))
Theorem	stgrclnbgr0 47950	All vertices of a star graph SN are in the closed neighborhood of the center. (Contributed by AV, 12-Sep-2025.)
⊢ 𝐺 = (StarGr‘𝑁)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐺 ClNeighbVtx 0) = 𝑉)
Theorem	isubgr3stgrlem1 47951	Lemma 1 for isubgr3stgr 47960. (Contributed by AV, 16-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝐹 = (𝐻 ∪ {⟨𝑋, 𝑌⟩})    ⇒   ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐹:𝐶–1-1-onto→(𝑅 ∪ {𝑌}))
Theorem	isubgr3stgrlem2 47952*	Lemma 2 for isubgr3stgr 47960. (Contributed by AV, 16-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))
Theorem	isubgr3stgrlem3 47953*	Lemma 3 for isubgr3stgr 47960. (Contributed by AV, 17-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0))
Theorem	isubgr3stgrlem4 47954*	Lemma 4 for isubgr3stgr 47960. (Contributed by AV, 24-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐴 = 𝑋 ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})
Theorem	isubgr3stgrlem5 47955*	Lemma 5 for isubgr3stgr 47960. (Contributed by AV, 24-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))    ⇒   ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌))
Theorem	isubgr3stgrlem6 47956*	Lemma 6 for isubgr3stgr 47960. (Contributed by AV, 24-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))    ⇒   ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐻:𝐼⟶(Edg‘(StarGr‘𝑁)))
Theorem	isubgr3stgrlem7 47957*	Lemma 7 for isubgr3stgr 47960. (Contributed by AV, 29-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))    ⇒   ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ 𝐽 ∈ (Edg‘(StarGr‘𝑁))) → (◡𝐹 “ 𝐽) ∈ 𝐼)
Theorem	isubgr3stgrlem8 47958*	Lemma 8 for isubgr3stgr 47960. (Contributed by AV, 29-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))    ⇒   ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐻:𝐼–1-1-onto→(Edg‘(StarGr‘𝑁)))
Theorem	isubgr3stgrlem9 47959*	Lemma 9 for isubgr3stgr 47960. (Contributed by AV, 29-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))    ⇒   ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐻:𝐼–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ 𝐼 (𝐹 “ 𝑒) = (𝐻‘𝑒)))
Theorem	isubgr3stgr 47960*	If a vertex of a simple graph has exactly 𝑁 (different) neighbors, and none of these neighbors are connected by an edge, then the (closed) neighborhood of this vertex induces a subgraph which is isomorphic to an 𝑁-star. (Contributed by AV, 29-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)    &   ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑆 = (StarGr‘𝑁)    &   ⊢ 𝑊 = (Vtx‘𝑆)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁)))
21.48.15.7  Local isomorphisms of graphs This section is about local isomorphisms of graphs, which are a generalization of isomorphisms of graphs, i.e., every isomorphism between two graphs is also a local isomorphism between these graphs, see uhgrimgrlim 47972. This definition is according to a chat in mathoverflow (https://mathoverflow.net/questions/491133/locally-isomorphic-graphs 47972): roughly speaking, it restricts the correspondence of two graphs to their neighborhoods. Additionally, a binary relation ≃𝑙𝑔𝑟 is defined (see df-grlic 47966) which is true for two graphs iff there is a local isomorphism between these graphs. Then these graphs are called "locally isomorphic". Therefore, this relation is also called "is locally isomorphic to" relation. As a main result of this section, it is shown that the "is locally isomorphic to" relation is an equivalence relation (for hypergraphs), see grlicer 48001. The names and symbols are chosen analogously to group isomorphisms GrpIso (see df-gim 19156) and graph isomorphisms GraphIso (see df-grim 47863) resp. isomorphism between groups ≃𝑔 (see df-gic 19157) and isomorphism between graphs ≃𝑔𝑟 (see df-gric 47866). As discussed in the above mentioned chat in mathoverflow, it is shown that there are local isomorphisms between two graphs which are not (ordinary) isomorphisms between these graphs. In other words, there are two different locally isomorphic graphs which are not isomorphic, see lgricngricex 48114. Such two graphs are the two generalized Petersen graphs G(5,K) of order 10 (see definition df-gpg 48026), which are the Petersen graph G(5,2) and the 5-prism G(5,1), see gpg5ngric 48113.
Syntax	cgrlim 47961	The class of graph local isomorphism sets.
class GraphLocIso
Syntax	cgrlic 47962	The class of the graph local isomorphism relation.
class ≃𝑙𝑔𝑟
Definition	df-grlim 47963*	A local isomorphism of graphs is a bijection between the sets of vertices of two graphs that preserves local adjacency, i.e. the subgraph induced by the closed neighborhood of a vertex of the first graph and the subgraph induced by the closed neighborhood of the associated vertex of the second graph are isomorphic. See the following chat in mathoverflow: https://mathoverflow.net/questions/491133/locally-isomorphic-graphs. (Contributed by AV, 27-Apr-2025.)
⊢ GraphLocIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))})
Theorem	grlimfn 47964	The graph local isomorphism function is a well-defined function. (Contributed by AV, 20-May-2025.)
⊢ GraphLocIso Fn (V × V)
Theorem	grlimdmrel 47965	The domain of the graph local isomorphism function is a relation. (Contributed by AV, 20-May-2025.)
⊢ Rel dom GraphLocIso
Definition	df-grlic 47966	Two graphs are said to be locally isomorphic iff they are connected by at least one local isomorphism. (Contributed by AV, 27-Apr-2025.)
⊢ ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o))
Theorem	isgrlim 47967*	A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. (Contributed by AV, 20-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))))
Theorem	isgrlim2 47968*	A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. Definitions expanded. (Contributed by AV, 29-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
Theorem	grlimprop 47969*	Properties of a local isomorphism of graphs. (Contributed by AV, 21-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣)))))
Theorem	grlimf1o 47970	A local isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 21-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)
Theorem	grlimprop2 47971*	Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
Theorem	uhgrimgrlim 47972	An isomorphism of hypergraphs is a local isomorphism between the two graphs. (Contributed by AV, 2-Jun-2025.)
⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
Theorem	uspgrlimlem1 47973*	Lemma 1 for uspgrlim 47977. (Contributed by AV, 16-Aug-2025.)
⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
Theorem	uspgrlimlem2 47974*	Lemma 2 for uspgrlim 47977. (Contributed by AV, 16-Aug-2025.)
⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (𝐻 ∈ USPGraph → (◡(iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀})
Theorem	uspgrlimlem3 47975*	Lemma 3 for uspgrlim 47977. (Contributed by AV, 16-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) → (𝑒 ∈ 𝐾 → (𝑓 “ 𝑒) = ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒)))
Theorem	uspgrlimlem4 47976*	Lemma 4 for uspgrlim 47977. (Contributed by AV, 16-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))
Theorem	uspgrlim 47977*	A local isomorphism of simple pseudographs is a bijection between their vertices that preserves neighborhoods, expressed by properties of their edges (not edge functions as in isgrlim2 47968). (Contributed by AV, 15-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))))))
Theorem	usgrlimprop 47978*	Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
Theorem	clnbgrvtxedg 47979*	An edge 𝐸 containing a vertex 𝐴 is an edge in the closed neighborhood of this vertex 𝐴. (Contributed by AV, 25-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐾)
Theorem	grlimedgclnbgr 47980*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there are two bijections 𝑓 and 𝑔 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴) and the edges between the vertices in 𝑁 onto the edges between the vertices in 𝑀, so that the mapped vertices of an edge 𝐸 containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 25-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸)))
Theorem	grlimprclnbgr 47981*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there are two bijections 𝑓 and 𝑔 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴) and the edges between the vertices in 𝑁 onto the edges between the vertices in 𝑀, so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 25-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))
Theorem	grlimprclnbgredg 47982*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 27-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
Theorem	grlimpredg 47983*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge in 𝐻. (Contributed by AV, 27-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽))
Theorem	grlimprclnbgrvtx 47984*	For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there is a bijection 𝑓 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀 containing the vertex (𝐹‘𝐴). (Contributed by AV, 28-Dec-2025.)
⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
Theorem	grlimgredgex 47985*	Local isomorphisms between simple pseudographs map an edge onto an edge with an endpoint being the image of one of the endpoints of the first edge under the local isomorphism. (Contributed by AV, 28-Dec-2025.)
⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝐼)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸)
Theorem	grlimgrtrilem1 47986*	Lemma 3 for grlimgrtri 47988. (Contributed by AV, 24-Aug-2025.) (Proof shortened by AV, 27-Dec-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))
Theorem	grlimgrtrilem2 47987*	Lemma 3 for grlimgrtri 47988. (Contributed by AV, 23-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   ⊢ 𝐼 = (Edg‘𝐺)    &   ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑎))    &   ⊢ 𝐽 = (Edg‘𝐻)    &   ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ⊢ (((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) ∧ {𝑏, 𝑐} ∈ 𝐾) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)
Theorem	grlimgrtri 47988*	If one of two locally isomorphic graphs has a triangle, so does the other. The triangle in the other graph is not necessarily the image (𝐹 “ 𝑇) of the triangle 𝑇 in the first graph. (Contributed by AV, 24-Aug-2025.)
⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))    ⇒   ⊢ (𝜑 → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐻))
Theorem	brgrlic 47989	The relation "is locally isomorphic to" for graphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
Theorem	brgrilci 47990	Prove that two graphs are locally isomorphic by an explicit local isomorphism. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐹 ∈ (𝑅 GraphLocIso 𝑆) → 𝑅 ≃𝑙𝑔𝑟 𝑆)
Theorem	grlicrel 47991	The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.)
⊢ Rel ≃𝑙𝑔𝑟
Theorem	grlicrcl 47992	Reverse closure of the "is locally isomorphic to" relation for graphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
Theorem	dfgrlic2 47993*	Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))
Theorem	grilcbri 47994*	Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣)))))
Theorem	dfgrlic3 47995*	Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣))    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
Theorem	grilcbri2 47996*	Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)    &   ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑋))    &   ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
Theorem	grlicref 47997	Graph local isomorphism is reflexive for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)
Theorem	grlicsym 47998	Graph local isomorphism is symmetric for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺))
Theorem	grlicsymb 47999	Graph local isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 9-Jun-2025.)
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑙𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑙𝑔𝑟 𝐴))
Theorem	grlictr 48000	Graph local isomorphism is transitive. (Contributed by AV, 10-Jun-2025.)
⊢ ((𝑅 ≃𝑙𝑔𝑟 𝑆 ∧ 𝑆 ≃𝑙𝑔𝑟 𝑇) → 𝑅 ≃𝑙𝑔𝑟 𝑇)