wellPowered_of_essentiallySmall_monoOver

English

If C is LocallySmall and WellPowered, then for any X, MonoOver X is EssentiallySmall.

Русский

Если C локально-малое и хорошо обеспечено, то для любого X MonoOver X = EssentiallySmall.

LaTeX

$$$\forall X,\text{EssentiallySmall}(\mathrm{MonoOver}(X)).$$$

Lean4

theorem wellPowered_of_essentiallySmall_monoOver [LocallySmall.{w} C] (h : ∀ X : C, EssentiallySmall.{w} (MonoOver X)) :
    WellPowered.{w} C :=
  { subobject_small := fun X => (essentiallySmall_monoOver_iff_small_subobject X).mp (h X) }

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