40 lines
1.0 KiB
Plaintext
40 lines
1.0 KiB
Plaintext
|
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
|