Theorems · Theorem · logic and foundations

FirstOrder.Language.Embedding.isElementary_of_exists

∀ {L : FirstOrder.Language} {M : Type u_1} {N : Type u_2} [inst : L.Structure M] [inst_1 : L.Structure N]
  (f : L.Embedding M N),
  (∀ (n : ℕ) (φ : L.BoundedFormula Empty (n + 1)) (x : Fin n → M) (a : N),
      φ.Realize default (Fin.snoc (⇑f ∘ x) a) → ∃ b, φ.Realize default (Fin.snoc (⇑f ∘ x) (f b))) →
    ∀ {n : ℕ} (φ : L.Formula (Fin n)) (x : Fin n → M), φ.Realize (⇑f ∘ x) ↔ φ.Realize x

The Tarski-Vaught test for elementarity of an embedding.

Defined in

Mathlib.ModelTheory.ElementaryMaps

Cited by

1 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FirstOrder.Language.StructureFirstOrder.Language.Structure