exists_of_card_le_finset — Mathlib

English

If α is finite and s ⊆ β is finite with |α| ≤ |s|, then there exists an embedding α ↪ β whose range is contained in s.

Русский

Пусть α конечно, s ⊆ β конечно и |α| ≤ |s|. Тогда существует вложение α ↪ β с образованием, contained в s.

LaTeX

$$$[Fintype\\alpha] \\{s: Finset\\beta\\}, \\ s.card \\ge |\\alpha| \\Rightarrow ∃ f:\\alpha \\hookrightarrow \\beta, \\operatorname{range}(f) \\subseteq s$$$

Lean4

theorem exists_of_card_le_finset [Fintype α] {s : Finset β} (h : Fintype.card α ≤ #s) : ∃ f : α ↪ β, Set.range f ⊆ s :=
  by
  rw [← Fintype.card_coe] at h
  rcases nonempty_of_card_le h with ⟨f⟩
  exact ⟨f.trans (Embedding.subtype _), by simp [Set.range_subset_iff]⟩

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