P. 478 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47701-47800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	clnbgrel 47701*	Characterization of a member 𝑁 of the closed neighborhood of a vertex 𝑋 in a graph 𝐺. (Contributed by AV, 9-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
Theorem	clnbgrvtxel 47702	Every vertex 𝐾 is a member of its closed neighborhood. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ (𝐺 ClNeighbVtx 𝐾))
Theorem	clnbgrisvtx 47703	Every member 𝑁 of the closed neighborhood of a vertex 𝐾 is a vertex. (Contributed by AV, 9-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑁 ∈ 𝑉)
Theorem	clnbgrssvtx 47704	The closed neighborhood of a vertex 𝐾 in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉
Theorem	clnbgrn0 47705	The closed neighborhood of a vertex is never empty. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)
Theorem	clnbupgr 47706*	The closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}))
Theorem	clnbupgrel 47707	A member of the closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ (𝑁 = 𝐾 ∨ {𝑁, 𝐾} ∈ 𝐸)))
Theorem	clnbgr0vtx 47708	In a null graph (with no vertices), all closed neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
⊢ ((Vtx‘𝐺) = ∅ → (𝐺 ClNeighbVtx 𝐾) = ∅)
Theorem	clnbgr0edg 47709	In an empty graph (with no edges), all closed neighborhoods consists of a single vertex. (Contributed by AV, 10-May-2025.)
⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾})
Theorem	clnbgrsym 47710	In a graph, the closed neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by AV, 10-May-2025.)
⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 ClNeighbVtx 𝑁))
Theorem	predgclnbgrel 47711	If a (not necessarily proper) unordered pair containing a vertex is an edge, the other vertex is in the closed neighborhood of the first vertex. (Contributed by AV, 23-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))
Theorem	clnbgredg 47712	A vertices connected by an edge with another vertex is a neigborhood of those vertex. (Contributed by AV, 24-Aug-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾)) → 𝑌 ∈ 𝑁)
Theorem	clnbgrssedg 47713	The vertices connected by an edge are a subset of the neigborhood of each of these vertices. (Contributed by AV, 26-May-2025.) (Proof shortened by AV, 24-Aug-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)
Theorem	edgusgrclnbfin 47714*	The size of the closed neighborhood of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐺 ClNeighbVtx 𝑈) ∈ Fin ↔ {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} ∈ Fin))
Theorem	clnbusgrfi 47715	The closed neighborhood of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ Fin ∧ 𝑈 ∈ 𝑉) → (𝐺 ClNeighbVtx 𝑈) ∈ Fin)
Theorem	clnbfiusgrfi 47716	The closed neighborhood of a vertex in a finite simple graph is a finite set. (Contributed by AV, 10-May-2025.)
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝑁) ∈ Fin)
Theorem	clnbgrlevtx 47717	The size of the closed neighborhood of a vertex is at most the number of vertices of a graph. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (♯‘(𝐺 ClNeighbVtx 𝑈)) ≤ (♯‘𝑉)
21.48.15.2  Semiclosed and semiopen neighborhoods (experimental) We have already definitions for open and closed neighborhoods of a vertex, which differs only in the fact that the first never contains the vertex, and the latter always contains the vertex. One of these definitions, however, cannot be simply derived from the other. This would be possible if a definition of a semiclosed neighborhood was available, see dfsclnbgr2 47718. The definitions for open and closed neighborhoods could be derived from such a more simple, but otherwise probably useless definition, see dfnbgr5 47723 and dfclnbgr5 47722. Depending on the existence of certain edges, a vertex belongs to its semiclosed neighborhood or not. An alternate approach is to introduce semiopen neighborhoods, see dfvopnbgr2 47725. The definitions for open and closed neighborhoods could also be derived from such a definition, see dfnbgr6 47729 and dfclnbgr6 47728. Like with semiclosed neighborhood, depending on the existence of certain edges, a vertex belongs to its semiopen neighborhood or not. It is unclear if either definition is/will be useful, and in contrast to dfsclnbgr2 47718, the definition of semiopen neighborhoods is much more complex.
Theorem	dfsclnbgr2 47718*	Alternate definition of the semiclosed neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiclosed neighborhood 𝑆 of a vertex 𝑁 is the set of all vertices incident with edges which join the vertex 𝑁 with a vertex. Therefore, a vertex is contained in its semiclosed neighborhood if it is connected with any vertex by an edge (see sclnbgrelself 47720), even only with itself (i.e., by a loop). (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
Theorem	sclnbgrel 47719*	Characterization of a member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
Theorem	sclnbgrelself 47720*	A vertex 𝑁 is a member of its semiclosed neighborhood iff there is an edge joining the vertex with a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))
Theorem	sclnbgrisvtx 47721*	Every member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 is a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝑉)
Theorem	dfclnbgr5 47722*	Alternate definition of the closed neighborhood of a vertex as union of the vertex with its semiclosed neighborhood. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑆))
Theorem	dfnbgr5 47723*	Alternate definition of the (open) neighborhood of a vertex as a semiclosed neighborhood without itself. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
Theorem	dfnbgrss 47724*	Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
Theorem	dfvopnbgr2 47725*	Alternate definition of the semiopen neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiopen neighborhood 𝑈 of a vertex 𝑁 is its open neighborhood together with itself if there is a loop at this vertex. (Contributed by AV, 15-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
Theorem	vopnbgrel 47726*	Characterization of a member 𝑋 of the semiopen neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))))
Theorem	vopnbgrelself 47727*	A vertex 𝑁 is a member of its semiopen neighborhood iff there is a loop joining the vertex with itself. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))
Theorem	dfclnbgr6 47728*	Alternate definition of the closed neighborhood of a vertex as union of the vertex with its semiopen neighborhood. (Contributed by AV, 17-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑈))
Theorem	dfnbgr6 47729*	Alternate definition of the (open) neighborhood of a vertex as a difference of its semiopen neighborhood and the singleton of itself. (Contributed by AV, 17-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
Theorem	dfsclnbgr6 47730*	Alternate definition of a semiclosed neighborhood of a vertex as a union of a semiopen neighborhood and the vertex itself if there is a loop at this vertex. (Contributed by AV, 17-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
Theorem	dfnbgrss2 47731*	Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    ⇒   ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
21.48.15.3  Induced subgraphs
Syntax	cisubgr 47732	Extend class notation with induced subgraphs.
class ISubGr
Definition	df-isubgr 47733*	Define the function mapping graphs and subsets of their vertices to their induced subgraphs. A subgraph induced by a subset of vertices of a graph is a subgraph of the graph which contains all edges of the graph that join vertices of the subgraph (see section I.1 in [Bollobas] p. 2 or section 1.1 in [Diestel] p. 4). Although a graph may be given in any meaningful representation, its induced subgraphs are always ordered pairs of vertices and edges. (Contributed by AV, 27-Apr-2025.)
⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
Theorem	isisubgr 47734*	The subgraph induced by a subset of vertices. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)
Theorem	isubgriedg 47735*	The edges of an induced subgraph. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))
Theorem	isubgrvtxuhgr 47736	The subgraph induced by the full set of vertices of a hypergraph. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)
Theorem	isubgrvtx 47737	The vertices of an induced subgraph. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
Theorem	isubgruhgr 47738	An induced subgraph of a hypergraph is a hypergraph. (Contributed by AV, 13-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)
Theorem	isubgrsubgr 47739	An induced subgraph of a hypergraph is a subgraph of the hypergraph. (Contributed by AV, 14-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)
Theorem	isubgrupgr 47740	An induced subgraph of a pseudograph is a pseudograph. (Contributed by AV, 14-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UPGraph)
Theorem	isubgrumgr 47741	An induced subgraph of a multigraph is a multigraph. (Contributed by AV, 15-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UMGraph)
Theorem	isubgrusgr 47742	An induced subgraph of a simple graph is a simple graph. (Contributed by AV, 15-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ USGraph)
Theorem	isubgr0uhgr 47743	The subgraph induced by an empty set of vertices of a hypergraph. (Contributed by AV, 13-May-2025.)
⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)
21.48.15.4  Isomorphisms of graphs This section is about isomorphisms of graphs, whereby the term "isomorphism" is used in both of its meanings (according to the Meriam-Webster dictionary, see https://www.merriam-webster.com/dictionary/isomorphism): "1: the quality or state of being isomorphic." and "2: a one-to-one correspondence between two mathematical sets". At first, an operation GraphIso is defined (see df-grim 47748) which provides the graph isomorphisms (as "one-to-one correspondence") between two given graphs. This definition, however, is applicable for any two sets, but is meaningful only if these sets have "vertices" and "edges". Afterwards, a binary relation ≃𝑔𝑟 is defined (see df-gric 47751) which is true for two graphs iff there is a graph isomorphisms between these graphs. Then these graphs are called "isomorphic". Therefore, this relation is also called "is isomorphic to" relation. More formally, 𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓𝑓 ∈ (𝐴 GraphIso 𝐵) resp. 𝐴 ≃𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅. Notice that there can be multiple isomorphisms between two graphs. For example, let ⟨{𝐴, 𝐵}, {{𝐴, 𝐵}}⟩ and ⟨{{𝑀, 𝑁}, {{𝑀, 𝑁}}⟩ be two graphs with two vertices and one edge, then 𝐴 ↦ 𝑀, 𝐵 ↦ 𝑁 and 𝐴 ↦ 𝑁, 𝐵 ↦ 𝑀 are two different isomorphisms between these graphs. The names and symbols are chosen analogously to group isomorphisms GrpIso (see df-gim 19299) resp. isomorphism between groups ≃𝑔 (see df-gic 19300). The general definition of graph isomorphisms and the relation "is isomorphic to" for graphs is specialized for simple hypergraphs (gricushgr 47770) and simple pseudographs (gricuspgr 47771). The latter corresponds to the definition in [Bollobas] p. 3. It is shown that the relation "is isomorphic to" for graphs is an equivalence relation, see gricer 47777. Finally, isomorphic graphs with different representations are studied (opstrgric 47779, ushggricedg 47780). Another approach could be to define a category of graphs (there are maybe multiple ones), where graph morphisms are couples consisting of a function on vertices and a function on edges with required compatibilities, as used in the definition of GraphIso. And then, a graph isomorphism is defined as an isomorphism in the category of graphs (something like "GraphIsom = ( Iso ` GraphCat )" ). Then general category theory theorems could be used, e.g., to show that graph isomorphism is an equivalence relation.
Syntax	cgrisom 47744	Extend class notation to include the graph ispmorphisms as pair.
class GraphIsom
Syntax	cgrim 47745	Extend class notation to include the graph ispmorphisms.
class GraphIso
Syntax	cgric 47746	Extend class notation to include the "is isomorphic to" relation for graphs.
class ≃𝑔𝑟
Definition	df-grisom 47747*	Define the class of all isomorphisms between two graphs. In contrast to (𝐹 GraphIso 𝐻), which is a set of functions between the vertices, (𝐹 GraphIsom 𝐻) is a set of pairs of functions: a function between the vertices, and a function between the (indices of the) edges. It is not clear if such a definition is useful. In the definition by [Diestel] p. 3, for example, the bijection between the vertices is called an isomorphism, as formalized in df-grim 47748. (Contributed by AV, 11-Dec-2022.) (New usage is discouraged.)
⊢ GraphIsom = (𝑥 ∈ V, 𝑦 ∈ V ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ 𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)))})
Definition	df-grim 47748*	An isomorphism between two graphs is a bijection between the sets of vertices of the two graphs that preserves adjacency, see definition in [Diestel] p. 3. (Contributed by AV, 19-Apr-2025.)
⊢ GraphIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))})
Theorem	grimfn 47749	The graph isomorphism function is a well-defined function. (Contributed by AV, 28-Apr-2025.)
⊢ GraphIso Fn (V × V)
Theorem	grimdmrel 47750	The domain of the graph isomorphism function is a relation. (Contributed by AV, 28-Apr-2025.)
⊢ Rel dom GraphIso
Definition	df-gric 47751	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 47752*	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 47753*	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 47754	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	isuspgrim0lem 47755*	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 47756*	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	uspgrimprop 47757*	An isomorphism of simple pseudographs 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. (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	isuspgrimlem 47758*	Lemma for isuspgrim 47759. (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 47759*	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	grimidvtxedg 47760	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 47761	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 47762	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 47763	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 47764	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	brgric 47765	The relation "is isomorphic to" for graphs. (Contributed by AV, 28-Apr-2025.)
⊢ (𝑅 ≃𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
Theorem	brgrici 47766	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 47767	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 47768*	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 47769*	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 47770*	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 47771*	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 47772	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 47773	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 47774	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 47775	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 47776	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 47777	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 47778	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 47779	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 47780	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	isubgrgrim 47781*	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 47782*	Lemma for uhgrimisgrgric 47783. (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 47783	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 47784*	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 47785*	Lemma for clnbgrgrim 47786: 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 47786	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 47787	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 47789 is designed for a special purpose, namely to provide a criterion for two graphs being not isomorphic (see grimgrtri 47798). 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 47789 may become obsolete.
Syntax	cgrtri 47788	Extend class notation with triangles (in a graph).
class GrTriangles
Definition	df-grtri 47789*	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 47796. (TODO: and a cycle of length 3 ,see grtricycl ). (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 47790	Lemma for grtriprop 47792. (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 47791*	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 47792*	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 47793	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 47794*	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 47795	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 47796	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	grtrimap 47797	Conditions for mapping triangles onto triangles. Lemma for grimgrtri 47798 and grlimgrtri 47820. (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 47798	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 47799*	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  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 47811. This definition is according to a chat in mathoverflow (https://mathoverflow.net/questions/491133/locally-isomorphic-graphs 47811): roughly speaking, it restricts the correspondence of two graphs to their neighborhoods. Additionally, a binary relation ≃𝑙𝑔𝑟 is defined (see df-grlic 47805) 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 47833. The names and symbols are chosen analogously to group isomorphisms GrpIso (see df-gim 19299) and graph isomorphisms GraphIso (see df-grim 47748) resp. isomorphism between groups ≃𝑔 (see df-gic 19300) and isomorphism between graphs ≃𝑔𝑟 (see df-gric 47751). In the future, it should be shown that there are local isomorphisms between two graphs which are not (ordinary) isomorphisms between these graphs, as dicussed in the above mentioned chat in mathoverflow.
Syntax	cgrlim 47800	The class of graph local isomorphism sets.
class GraphLocIso