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Theorems · Theorem · order theory

InitialSeg.lt_apply_iff

∀ {α : Type u_1} {β : Type u_2} [inst : PartialOrder β] {a : α} {b : β} [inst_1 : PartialOrder α] (f : α ≤i β),
  b < f a ↔ ∃ a' < a, f a' = b
Defined in
Mathlib.Order.InitialSeg
Cited by
3 results in Mathlib
Foundations
Depth 18 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
PartialOrderPartialOrder
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Ordinal.lt_lift_iff · cited by 4Ordinal.lt_lift_iffCardinal.lt_lift_iff · cited by 4Cardinal.lt_lift_iffPrincipalSeg.lt_apply_iff · cited by 0PrincipalSeg.lt_apply_iffDFunLike.coe · cited by 62936DFunLike.coePartialOrder · cited by 6410PartialOrderInitialSeg · cited by 70InitialSegInitialSeg.mem_range_of_rel · cited by 6InitialSeg.mem_range_of_r…InitialSeg.strictMono · cited by 4InitialSeg.strictMonoInitialSeg.lt_iff_lt · cited by 3InitialSeg.lt_iff_ltInitialSeg.lt_apply_iffCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by3

Results whose statement or proof uses this declaration.