bookshelf/Bookshelf/Enderton/Set/Chapter_6.lean

40 lines
1.0 KiB
Plaintext
Raw Blame History

This file contains ambiguous Unicode characters!

This file contains ambiguous Unicode characters that may be confused with others in your current locale. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to highlight these characters.

import Mathlib.Data.Set.Function
import Mathlib.Data.Rel
/-! # Enderton.Set.Chapter_6
Cardinal Numbers and the Axiom of Choice
-/
namespace Enderton.Set.Chapter_6
/-! #### Theorem 6A
For any sets `A`, `B`, and `C`,
(a) `A ≈ A`.
(b) If `A ≈ B`, then `B ≈ A`.
(c) If `A ≈ B` and `B ≈ C`, then `A ≈ C`.
-/
theorem theorem_6a_a (A : Set α)
: ∃ f, Set.BijOn f A A := by
refine ⟨fun x => x, ?_⟩
unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
simp only [imp_self, implies_true, Set.image_id', true_and]
exact Eq.subset rfl
theorem theorem_6a_b [Nonempty α] (A : Set α) (B : Set β)
(f : α → β) (hf : Set.BijOn f A B)
: ∃ g, Set.BijOn g B A := by
refine ⟨Function.invFunOn f A, ?_⟩
exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hf).mpr hf
theorem theorem_6a_c (A : Set α) (B : Set β) (C : Set γ)
(f : α → β) (hf : Set.BijOn f A B)
(g : β → γ) (hg : Set.BijOn g B C)
: ∃ h, Set.BijOn h A C := by
exact ⟨g ∘ f, Set.BijOn.comp hg hf⟩
end Enderton.Set.Chapter_6