biUnion_congr — Mathlib

English

If π1.boxes ⊆ π2.boxes and π2.iUnion ⊆ π1.iUnion, then π1 = π2.

Русский

Если box-e π1_subset π2 и iUnion π2 ⊆ iUnion π1, то π1 = π2.

LaTeX

$$π₁.boxes ⊆ π₂.boxes → π₂.iUnion ⊆ π₁.iUnion → π₁ = π₂$$

Lean4

@[congr]
theorem biUnion_congr (h : π₁ = π₂) (hi : ∀ J ∈ π₁, πi₁ J = πi₂ J) : π₁.biUnion πi₁ = π₂.biUnion πi₂ :=
  by
  subst π₂
  ext J
  simp only [mem_biUnion]
  constructor <;> exact fun ⟨J', h₁, h₂⟩ => ⟨J', h₁, hi J' h₁ ▸ h₂⟩

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