English
The unconditional filter is preserved by comaps along embeddings: (unconditional β).comap f = unconditional γ.
Русский
Безусловный фильтр сохраняется при комап沿 вложения: (unconditional β).comap f = unconditional γ.
LaTeX
$$$ (unconditional\\,\\beta).comap f = unconditional\\,\\gamma $$$
Lean4
/-- The unconditional filter is preserved by comaps. -/
@[simp]
theorem comap_unconditional {β} (f : γ ↪ β) : (unconditional β).comap f = unconditional γ := by
classical
simp only [unconditional, comap]
congr 1 with s
simp only [mem_map, mem_atTop_sets, ge_iff_le, Finset.le_eq_subset, mem_preimage]
constructor <;> rintro ⟨t, ht⟩
· refine ⟨t.preimage f (by simp), fun x hx ↦ ?_⟩
simpa [Finset.union_eq_right.mpr hx] using ht (t ∪ x.map f) t.subset_union_left
· exact ⟨_, fun b hb ↦ ht _ (Finset.map_subset_iff_subset_preimage.mp hb)⟩
