Theorems · Theorem · order theory
Monotone.mem_upperBounds_image
∀ {α : Type u} {β : Type v} [inst : Preorder α] [inst_1 : Preorder β] {f : α → β},
Monotone f → ∀ {a : α} {s : Set α}, a ∈ upperBounds s → f a ∈ upperBounds (f '' s)- Defined in
- Mathlib.Order.Bounds.Image
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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.imagestatement · cited by 5,609
- Monotonestatement and proof · cited by 1,397
- upperBoundsstatement and proof · cited by 263
- Set.forall_mem_imageproof · cited by 65
Cited by12
Results whose statement or proof uses this declaration.
- Monotone.map_bddAboveproof · cited by 9
- GaloisConnection.isLUB_l_imageproof · cited by 5
- Monotone.map_isGreatestproof · cited by 4
- Monotone.image_upperBounds_subset_upperBounds_imageproof · cited by 3
- Monotone.csSup_image_leproof · cited by 2
- isLUB_prodproof · cited by 2
- Topology.IsScott.scottContinuousOn_iff_continuousproof · cited by 2
- IsLUB.of_imageproof · cited by 1
- GaloisInsertion.isLUB_of_u_imageproof · cited by 0
- Antitone.mem_lowerBounds_imageproof · cited by 0
- GaloisCoinsertion.isLUB_of_l_imageproof · cited by 0
- isLUB_piproof · cited by 0