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	upgrimwlklem5 47901*	Lemma 5 for upgrimwlk 47902. (Contributed by AV, 28-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    ⇒   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐸))) → (𝑁 “ (𝐼‘(𝐹‘𝑖))) = {((𝑁 ∘ 𝑃)‘𝑖), ((𝑁 ∘ 𝑃)‘(𝑖 + 1))})
Theorem	upgrimwlk 47902*	Graph isomorphisms between simple pseudographs map walks onto walks. (Contributed by AV, 28-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → 𝐸(Walks‘𝐻)(𝑁 ∘ 𝑃))
Theorem	upgrimwlklen 47903*	Graph isomorphisms between simple pseudographs map walks onto walks of the same length. (Contributed by AV, 6-Nov-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → (𝐸(Walks‘𝐻)(𝑁 ∘ 𝑃) ∧ (♯‘𝐸) = (♯‘𝐹)))
Theorem	upgrimtrlslem1 47904*	Lemma 1 for upgrimtrls 47906. (Contributed by AV, 29-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ dom 𝐹) → (𝑁 “ (𝐼‘(𝐹‘𝑋))) ∈ (Edg‘𝐻))
Theorem	upgrimtrlslem2 47905*	Lemma 2 for upgrimtrls 47906. (Contributed by AV, 29-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹)) → ((◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))) = (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑦)))) → 𝑥 = 𝑦))
Theorem	upgrimtrls 47906*	Graph isomorphisms between simple pseudographs map trails onto trails. (Contributed by AV, 29-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → 𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃))
Theorem	upgrimpthslem1 47907*	Lemma 1 for upgrimpths 47909. (Contributed by AV, 30-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → Fun ◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
Theorem	upgrimpthslem2 47908*	Lemma 2 for upgrimpths 47909. (Contributed by AV, 31-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
Theorem	upgrimpths 47909*	Graph isomorphisms between simple pseudographs map paths onto paths. (Contributed by AV, 31-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))
Theorem	upgrimspths 47910*	Graph isomorphisms between simple pseudographs map simple paths onto simple paths. (Contributed by AV, 31-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → 𝐸(SPaths‘𝐻)(𝑁 ∘ 𝑃))
Theorem	upgrimcycls 47911*	Graph isomorphisms between simple pseudographs map cycles onto cycles. (Contributed by AV, 31-Oct-2025.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))    &   ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))    &   ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → 𝐸(Cycles‘𝐻)(𝑁 ∘ 𝑃))
Theorem	brgric 47912	The relation "is isomorphic to" for graphs. (Contributed by AV, 28-Apr-2025.)
⊢ (𝑅 ≃𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
Theorem	brgrici 47913	Prove that two graphs are isomorphic by an explicit isomorphism. (Contributed by AV, 28-Apr-2025.)
⊢ (𝐹 ∈ (𝑅 GraphIso 𝑆) → 𝑅 ≃𝑔𝑟 𝑆)
Theorem	gricrcl 47914	Reverse closure of the "is isomorphic to" relation for graphs. (Contributed by AV, 12-Jun-2025.)
⊢ (𝐺 ≃𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
Theorem	dfgric2 47915*	Alternate, explicit definition of the "is isomorphic to" relation for two graphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)
⊢ 𝑉 = (Vtx‘𝐴)    &   ⊢ 𝑊 = (Vtx‘𝐵)    &   ⊢ 𝐼 = (iEdg‘𝐴)    &   ⊢ 𝐽 = (iEdg‘𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (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	gricbri 47916*	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 47917*	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 47918*	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 47919	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 47920	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 47921	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 47922	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 47923	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 47924	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 47925	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 47926	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 47927	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 47928*	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 47929*	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 47930*	Lemma for uhgrimisgrgric 47931. (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 47931	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 47932*	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 47933*	Lemma for clnbgrgrim 47934: 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 47934	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 47935	Graph isomorphisms map edges onto the corresponding edges. (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)))
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 47937 is designed for a special purpose, namely to provide a criterion for two graphs being not isomorphic (see grimgrtri 47948). 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 47937 may become obsolete.
Syntax	cgrtri 47936	Extend class notation with triangles (in a graph).
class GrTriangles
Definition	df-grtri 47937*	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 47944, and the vertices of a cycle of size 3 are a triangle in a graph, see cycl3grtri 47946. (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 47938	Lemma for grtriprop 47940. (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 47939*	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 47940*	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 47941	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 47942*	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 47943	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 47944	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 47945	Lemma for cycl3grtri 47946. (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 47946	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 47947	Conditions for mapping triangles onto triangles. Lemma for grimgrtri 47948 and grlimgrtri 47995. (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 47948	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 47949*	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 47950	Extend class notation with star graphs.
class StarGr
Definition	df-stgr 47951*	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 47959, (StarGr‘1) will be a single edge (graph with two vertices connected by an edge), see stgr1 47960, 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 47952*	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 47953	The vertices of the star graph SN. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
Theorem	stgriedg 47954*	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 47955*	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 47956*	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 47957*	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 47958	The star graph SN is a simple graph. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
Theorem	stgr0 47959	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 47960	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 47961	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 47962	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 47963	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 47964	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 47965	Lemma 1 for isubgr3stgr 47974. (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 47966*	Lemma 2 for isubgr3stgr 47974. (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 47967*	Lemma 3 for isubgr3stgr 47974. (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 47968*	Lemma 4 for isubgr3stgr 47974. (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 47969*	Lemma 5 for isubgr3stgr 47974. (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 47970*	Lemma 6 for isubgr3stgr 47974. (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 47971*	Lemma 7 for isubgr3stgr 47974. (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 47972*	Lemma 8 for isubgr3stgr 47974. (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 47973*	Lemma 9 for isubgr3stgr 47974. (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 47974*	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 47986. This definition is according to a chat in mathoverflow (https://mathoverflow.net/questions/491133/locally-isomorphic-graphs 47986): roughly speaking, it restricts the correspondence of two graphs to their neighborhoods. Additionally, a binary relation ≃𝑙𝑔𝑟 is defined (see df-grlic 47980) 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 48008. The names and symbols are chosen analogously to group isomorphisms GrpIso (see df-gim 19191) and graph isomorphisms GraphIso (see df-grim 47878) resp. isomorphism between groups ≃𝑔 (see df-gic 19192) and isomorphism between graphs ≃𝑔𝑟 (see df-gric 47881). 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 48119. Such two graphs are the two generalized Petersen graphs G(5,K) of order 10 (see definition df-gpg 48032), which are the Petersen graph G(5,2) and the 5-prism G(5,1), see gpg5ngric 48118.
Syntax	cgrlim 47975	The class of graph local isomorphism sets.
class GraphLocIso
Syntax	cgrlic 47976	The class of the graph local isomorphism relation.
class ≃𝑙𝑔𝑟
Definition	df-grlim 47977*	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 47978	The graph local isomorphism function is a well-defined function. (Contributed by AV, 20-May-2025.)
⊢ GraphLocIso Fn (V × V)
Theorem	grlimdmrel 47979	The domain of the graph local isomorphism function is a relation. (Contributed by AV, 20-May-2025.)
⊢ Rel dom GraphLocIso
Definition	df-grlic 47980	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 47981*	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 47982*	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 47983*	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 47984	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 47985*	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 47986	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 47987*	Lemma 1 for uspgrlim 47991. (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 47988*	Lemma 2 for uspgrlim 47991. (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 47989*	Lemma 3 for uspgrlim 47991. (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 47990*	Lemma 4 for uspgrlim 47991. (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 47991*	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 47982). (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 47992*	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	grlimgrtrilem1 47993*	Lemma 3 for grlimgrtri 47995. (Contributed by AV, 24-Aug-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 47994*	Lemma 3 for grlimgrtri 47995. (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 47995*	Local isomorphisms between simple pseudographs map triangles onto triangles. (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 47996	The relation "is locally isomorphic to" for graphs. (Contributed by AV, 9-Jun-2025.)
⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
Theorem	brgrilci 47997	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 47998	The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.)
⊢ Rel ≃𝑙𝑔𝑟
Theorem	grlicrcl 47999	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 48000*	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))))))