image_subset_iff — Mathlib

English

The image operator and subset relation form a Galois connection: f''S ⊆ T iff S ⊆ f^{-1}(T).

Русский

Оператор образа и отношение подмножества образуют связь Галлиса: f''S ⊆ T ⇔ S ⊆ f^{-1}(T).

LaTeX

$$$$ f''S \subseteq T \iff S \subseteq f^{-1}(T). $$$$

Lean4

/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
  forall_mem_image

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