Theorems · Theorem · general topology

Filter.map_piMap_pi

∀ {ι : Type u_1} {α : ι → Type u_4} {β : ι → Type u_5} {f : (i : ι) → α i → β i},
  (∀ᶠ (i : ι) in Filter.cofinite, Function.Surjective (f i)) →
    ∀ (l : (i : ι) → Filter (α i)), Filter.map (Pi.map f) (Filter.pi l) = Filter.pi fun i => Filter.map (f i) (l i)

Given a family of maps f i : α i → β i and a family of filters l i : Filter (α i), if all but finitely many of f i are surjective, then the indexed product of f is maps the indexed product of the filters l i to the indexed products of their pushforwards under individual f is. See also map_piMap_pi_finite for the case of a finite index type.

Defined in

Mathlib.Order.Filter.Cofinite

Cited by

2 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound