Theorems · Theorem · order theory
DirectedOn.le_ciInf_set_iff
∀ {α : Type u_1} [inst : ConditionallyCompletePartialOrderInf α] {ι : Type u_5} {s : Set ι} {f : ι → α} {a : α},
s.Nonempty → DirectedOn (fun x1 x2 => x2 ≤ x1) (f '' s) → BddBelow (f '' s) → (a ≤ ⨅ i, f ↑i ↔ ∀ i ∈ s, a ≤ f i)- Cited by
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- Foundations
- Depth 10 from the axioms · uses propext, Quot.sound
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Cites11
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
- Set.Elemstatement · cited by 7,166
- Set.imagestatement and proof · cited by 5,609
- Set.Nonemptystatement and proof · cited by 2,627
- iInfstatement · cited by 1,690
- BddBelowstatement and proof · cited by 401
- DirectedOnstatement and proof · cited by 271
- Set.forall_mem_imageproof · cited by 65
- ConditionallyCompletePartialOrderInfstatement and proof · cited by 51
- le_isGLB_iffproof · cited by 17
- DirectedOn.isGLB_ciInf_setproof · cited by 1
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