Theorems · Theorem · logic and foundations

FirstOrder.Language.DirectLimit.range_lift

∀ {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] {P : Type u₁} [inst_5 : L.Structure P]
  (g : (i : ι) → L.Embedding (G i) P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x),
  (FirstOrder.Language.DirectLimit.lift L ι G f g Hg).toHom.range = ⨆ i, (g i).toHom.range

Defined in

Mathlib.ModelTheory.DirectLimit

Cited by

1 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFirstOrder.Language.StructureIsDirectedOrderDirectedSystemNonemptyFirstOrder.Language.Structure