biUnion_diff_biUnion_subset — Mathlib

English

For a two-parameter family f(i,j), Iic of iInf equals the double intersection of Iic(f(i,j)).

Русский

Для двойного семейства f(i,j) Iic(iInf f) равно двойному пересечению Iic(f(i,j)).

LaTeX

$$$$ \\mathrm{Iic}(\\mathrm{iInf}_{i,j} f(i,j)) = \\bigcap_i \\bigcap_j \\mathrm{Iic}(f(i,j)). $$$$

Lean4

theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) : ((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x :=
  by
  simp only [diff_subset_iff, ← biUnion_union]
  apply biUnion_subset_biUnion_left
  rw [union_diff_self]
  apply subset_union_right

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