preimage_comp — Mathlib

English

Equality of realizations is preserved under relabeling in the preimage construction.

Русский

Реализации сохраняют равенство под переименованием в конструкцию предобраза.

LaTeX

$$$Eq ((FirstOrder.Language.Formula.relabel g φ).Realize v) (φ.Realize (Function.comp v g)).$$$

Lean4

theorem preimage_comp (f : α → β) {s : Set (α → M)} (h : A.Definable L s) :
    A.Definable L ((fun g : β → M => g ∘ f) ⁻¹' s) :=
  by
  obtain ⟨φ, rfl⟩ := h
  refine ⟨φ.relabel f, ?_⟩
  ext
  simp only [Set.preimage_setOf_eq, mem_setOf_eq, Formula.realize_relabel]

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