map — Mathlib

English

For any graph hom f, comap f ⊤ = ⊤.

Русский

Для любого отображения f: G → H, comap f верхний подграф равен верхнему.

LaTeX

$$$ Subgraph.comap f \top = \top $$$

Lean4

/-- Graph homomorphisms induce a covariant function on subgraphs. -/
@[simps]
protected def map (f : G →g G') (H : G.Subgraph) : G'.Subgraph
    where
  verts := f '' H.verts
  Adj := Relation.Map H.Adj f f
  adj_sub := by
    rintro _ _ ⟨u, v, h, rfl, rfl⟩
    exact f.map_rel (H.adj_sub h)
  edge_vert := by
    rintro _ _ ⟨u, v, h, rfl, rfl⟩
    exact Set.mem_image_of_mem _ (H.edge_vert h)
  symm := by
    rintro _ _ ⟨u, v, h, rfl, rfl⟩
    exact ⟨v, u, H.symm h, rfl, rfl⟩

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