Theorems · Theorem · order theory
isLUB_iUnion_iff_of_isLUB
∀ {α : Type u_1} [inst : Preorder α] {ι : Sort u_2} {s : ι → Set α} {u : ι → α},
(∀ (i : ι), IsLUB (s i) (u i)) → ∀ (c : α), IsLUB (Set.range u) c ↔ IsLUB (⋃ i, s i) c- Defined in
- Mathlib.Order.Bounds.Lattice
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Preorderstatement and proof · cited by 7,952
- Set.rangestatement · cited by 4,705
- Set.iUnionstatement and proof · cited by 2,483
- Set.iInterproof · cited by 1,084
- Set.Iciproof · cited by 1,070
- IsLUBstatement and proof · cited by 280
- upperBoundsproof · cited by 263
- IsLUB.upperBounds_eqproof · cited by 10
- Set.range_eq_iUnionproof · cited by 4
- upperBounds_singletonproof · cited by 4
- isLUB_congrproof · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- ScottContinuousOn.fromProdproof · cited by 1