Theorems · Theorem · logic and foundations

FirstOrder.Language.DirectLimit.exists_fg_substructure_in_Sigma

∀ {L : FirstOrder.Language} {ι : Type v} [inst : Preorder ι] {G : ι → Type w} [inst_1 : (i : ι) → L.Structure (G i)]
  {f : (i j : ι) → i ≤ j → L.Embedding (G i) (G j)} [inst_2 : IsDirectedOrder ι]
  [inst_3 : DirectedSystem G fun i j h => ⇑(f i j h)] [inst_4 : Nonempty ι]
  (S : L.Substructure (FirstOrder.Language.DirectLimit G f)),
  S.FG → ∃ i T, FirstOrder.Language.Substructure.map (FirstOrder.Language.DirectLimit.of L ι G f i).toHom T = S

Every finitely generated substructure of the direct limit corresponds to some substructure in some component of the directed system.

Defined in

Mathlib.ModelTheory.DirectLimit

Cited by

0 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFirstOrder.Language.StructureIsDirectedOrderDirectedSystemNonempty