Theorems · Theorem · logic and foundations

FirstOrder.Language.DirectLimit.exists_unify_eq

∀ {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)] [Nonempty ι] {α : Type u_1} [Finite α]
  {x y : α → FirstOrder.Language.Structure.Sigma f},
  x ≈ y →
    ∃ i,
      ∃ (hx : i ∈ upperBounds (Set.range (Sigma.fst ∘ x))) (hy : i ∈ upperBounds (Set.range (Sigma.fst ∘ y))),
        FirstOrder.Language.DirectLimit.unify f x i hx = FirstOrder.Language.DirectLimit.unify f y i hy

Defined in

Mathlib.ModelTheory.DirectLimit

Cited by

0 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFirstOrder.Language.StructureIsDirectedOrderDirectedSystemNonemptyFinite