P. 486 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48501-48600   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-gric 48501	Two graphs are said to be isomorphic iff they are connected by at least one isomorphism, see definition in [Diestel] p. 3 and definition in [Bollobas] p. 3. Isomorphic graphs share all global graph properties like order and size. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 19-Apr-2025.)
⊢ ≃𝑔𝑟 = (◡ GraphIso “ (V ∖ 1o))
Theorem	isgrim 48502*	An isomorphism of graphs is a bijection between their vertices that preserves adjacency. (Contributed by AV, 19-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (iEdg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))
Theorem	grimprop 48503*	Properties of an isomorphism of graphs. (Contributed by AV, 29-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (iEdg‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))))
Theorem	grimf1o 48504	An isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 29-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)
Theorem	grimidvtxedg 48505	The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and a graph with the same vertices and edges. (Contributed by AV, 4-May-2025.)
⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))    &   ⊢ (𝜑 → (iEdg‘𝐺) = (iEdg‘𝐻))    ⇒   ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))
Theorem	grimid 48506	The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and itself. (Contributed by AV, 29-Apr-2025.) (Prove shortened by AV, 5-May-2025.)
⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺))
Theorem	grimuhgr 48507	If there is a graph isomorphism between a hypergraph and a class with an edge function, the class is also a hypergraph. (Contributed by AV, 2-May-2025.)
⊢ ((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇) ∧ Fun (iEdg‘𝑇)) → 𝑇 ∈ UHGraph)
Theorem	grimcnv 48508	The converse of a graph isomorphism is a graph isomorphism. (Contributed by AV, 1-May-2025.)
⊢ (𝑆 ∈ UHGraph → (𝐹 ∈ (𝑆 GraphIso 𝑇) → ◡𝐹 ∈ (𝑇 GraphIso 𝑆)))
Theorem	grimco 48509	The composition of graph isomorphisms is a graph isomorphism. (Contributed by AV, 3-May-2025.)
⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GraphIso 𝑈))
Theorem	uhgrimedgi 48510	An isomorphism between graphs preserves edges, i.e. if there is an edge in one graph connecting vertices then there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (Edg‘𝐻)    ⇒   ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)
Theorem	uhgrimedg 48511	An isomorphism between graphs preserves edges, i.e. there is an edge in one graph connecting vertices iff there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (Edg‘𝐻)    ⇒   ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐾 ∈ 𝐸 ↔ (𝐹 “ 𝐾) ∈ 𝐷))
Theorem	uhgrimprop 48512*	An isomorphism between hypergraphs is a bijection between their vertices that preserves adjacency for simple edges, i.e. there is a simple edge in one graph connecting one or two vertices iff there is a simple edge in the other graph connecting the vertices which are the images of the vertices. (Contributed by AV, 27-Apr-2025.) (Revised by AV, 25-Oct-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (Edg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)))
Theorem	isuspgrim0lem 48513*	An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (Edg‘𝐻)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑀 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))    &   ⊢ 𝑁 = (𝑥 ∈ dom 𝐼 ↦ (◡𝐽‘(𝑀‘(𝐼‘𝑥))))    ⇒   ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑀:𝐸–1-1-onto→𝐷) → (𝑁:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑁‘𝑖)) = (𝐹 “ (𝐼‘𝑖))))
Theorem	isuspgrim0 48514*	An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))
Theorem	isuspgrimlem 48515*	Lemma for isuspgrim 48516. (Contributed by AV, 27-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (Edg‘𝐻)    ⇒   ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)
Theorem	isuspgrim 48516*	A class is an isomorphism of simple pseudographs iff it is a bijection between their vertices that preserves adjacency, i.e. there is an edge in one graph connecting one or two vertices iff there is an edge in the other graph connecting the vertices which are the images of the vertices. This corresponds to the formal definition in [Bollobas] p. 3 and the definition in [Diestel] p. 3. (Contributed by AV, 27-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))
Theorem	upgrimwlklem1 48517*	Lemma 1 for upgrimwlk 48522 and upgrimwlklen 48523. (Contributed by AV, 25-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 ∈ Word dom 𝐼)    ⇒   ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐹))
Theorem	upgrimwlklem2 48518*	Lemma 2 for upgrimwlk 48522. (Contributed by AV, 25-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 ∈ Word dom 𝐼)    ⇒   ⊢ (𝜑 → 𝐸 ∈ Word dom 𝐽)
Theorem	upgrimwlklem3 48519*	Lemma 3 for upgrimwlk 48522. (Contributed by AV, 25-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 ∈ Word dom 𝐼)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0..^(♯‘𝐸))) → (𝐽‘(𝐸‘𝑋)) = (𝑁 “ (𝐼‘(𝐹‘𝑋))))
Theorem	upgrimwlklem4 48520*	Lemma 4 for upgrimwlk 48522. (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 ∈ Word dom 𝐼)    &   ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))    ⇒   ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻))
Theorem	upgrimwlklem5 48521*	Lemma 5 for upgrimwlk 48522. (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 48522*	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 48523*	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 48524*	Lemma 1 for upgrimtrls 48526. (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 48525*	Lemma 2 for upgrimtrls 48526. (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 48526*	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 48527*	Lemma 1 for upgrimpths 48529. (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 48528*	Lemma 2 for upgrimpths 48529. (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 48529*	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 48530*	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 48531*	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 48532	The relation "is isomorphic to" for graphs. (Contributed by AV, 28-Apr-2025.)
⊢ (𝑅 ≃𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
Theorem	brgrici 48533	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 48534	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 48535*	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 48536*	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 48537*	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 48538*	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 48539	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 48540	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 48541	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 48542	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 48543	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 48544	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 48545	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 48546	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 48547	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 48548*	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 48549*	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 48550*	Lemma for uhgrimisgrgric 48551. (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 48551	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 48552*	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 48553*	Lemma for clnbgrgrim 48554: 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 48554	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 48555	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 48556	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 48558 is designed for a special purpose, namely to provide a criterion for two graphs being not isomorphic (see grimgrtri 48569). 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 48558 may become obsolete.
Syntax	cgrtri 48557	Extend class notation with triangles (in a graph).
class GrTriangles
Definition	df-grtri 48558*	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 48565, and the vertices of a cycle of size 3 are a triangle in a graph, see cycl3grtri 48567. (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 48559	Lemma for grtriprop 48561. (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 48560*	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 48561*	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 48562	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 48563*	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 48564	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 48565	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 48566	Lemma for cycl3grtri 48567. (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 48567	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 48568	Conditions for mapping triangles onto triangles. Lemma for grimgrtri 48569 and grlimgrtri 48623. (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 48569	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 48570*	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 48571	Extend class notation with star graphs.
class StarGr
Definition	df-stgr 48572*	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 48580, (StarGr‘1) will be a single edge (graph with two vertices connected by an edge), see stgr1 48581, 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 48573*	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 48574	The vertices of the star graph SN. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
Theorem	stgriedg 48575*	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 48576*	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 48577*	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 48578*	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 48579	The star graph SN is a simple graph. (Contributed by AV, 11-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
Theorem	stgr0 48580	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 48581	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 48582	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 48583	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 48584	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 48585	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 48586	Lemma 1 for isubgr3stgr 48595. (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 48587*	Lemma 2 for isubgr3stgr 48595. (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 48588*	Lemma 3 for isubgr3stgr 48595. (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 48589*	Lemma 4 for isubgr3stgr 48595. (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 48590*	Lemma 5 for isubgr3stgr 48595. (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 48591*	Lemma 6 for isubgr3stgr 48595. (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 48592*	Lemma 7 for isubgr3stgr 48595. (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 48593*	Lemma 8 for isubgr3stgr 48595. (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 48594*	Lemma 9 for isubgr3stgr 48595. (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 48595*	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 48607. This definition is according to a chat in mathoverflow (https://mathoverflow.net/questions/491133/locally-isomorphic-graphs 48607): roughly speaking, it restricts the correspondence of two graphs to their neighborhoods. Additionally, a binary relation ≃𝑙𝑔𝑟 is defined (see df-grlic 48601) 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 48636. The names and symbols are chosen analogously to group isomorphisms GrpIso (see df-gim 19320) and graph isomorphisms GraphIso (see df-grim 48498) resp. isomorphism between groups ≃𝑔 (see df-gic 19321) and isomorphism between graphs ≃𝑔𝑟 (see df-gric 48501). 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 48749. Such two graphs are the two generalized Petersen graphs G(5,K) of order 10 (see definition df-gpg 48661), which are the Petersen graph G(5,2) and the 5-prism G(5,1), see gpg5ngric 48748.
Syntax	cgrlim 48596	The class of graph local isomorphism sets.
class GraphLocIso
Syntax	cgrlic 48597	The class of the graph local isomorphism relation.
class ≃𝑙𝑔𝑟
Definition	df-grlim 48598*	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 48599	The graph local isomorphism function is a well-defined function. (Contributed by AV, 20-May-2025.)
⊢ GraphLocIso Fn (V × V)
Theorem	grlimdmrel 48600	The domain of the graph local isomorphism function is a relation. (Contributed by AV, 20-May-2025.)
⊢ Rel dom GraphLocIso