50 lines
1.2 KiB
Text
50 lines
1.2 KiB
Text
import data.set
|
|
namespace function
|
|
section
|
|
open set
|
|
variables {A B : Type}
|
|
set_option pp.beta false
|
|
definition bijective (f : A → B) := injective f ∧ surjective f
|
|
|
|
lemma injective_eq_inj_on_univ₁ (f : A → B) : injective f = inj_on f univ :=
|
|
begin
|
|
esimp [injective, inj_on, univ, mem],
|
|
apply propext,
|
|
apply iff.intro,
|
|
intro Pl a1 a2,
|
|
rewrite *true_imp,
|
|
exact Pl a1 a2,
|
|
intro Pr a1 a2,
|
|
exact Pr trivial trivial
|
|
end
|
|
|
|
lemma injective_eq_inj_on_univ₂ (f : A → B) : injective f = inj_on f univ :=
|
|
begin
|
|
esimp [injective, inj_on, univ, mem],
|
|
apply propext,
|
|
apply iff.intro,
|
|
intro Pl a1 a2,
|
|
rewrite *(propext !true_imp),
|
|
exact Pl a1 a2,
|
|
intro Pr a1 a2,
|
|
exact Pr trivial trivial
|
|
end
|
|
|
|
lemma injective_eq_inj_on_univ₃ (f : A → B) : injective f = inj_on f univ :=
|
|
begin
|
|
esimp [injective, inj_on, univ, mem],
|
|
apply propext,
|
|
repeat (apply forall_congr; intros),
|
|
rewrite *(propext !true_imp)
|
|
end
|
|
|
|
lemma injective_eq_inj_on_univ₄ (f : A → B) : injective f = inj_on f univ :=
|
|
begin
|
|
esimp [injective, inj_on, univ, mem],
|
|
apply propext,
|
|
repeat (apply forall_congr; intros),
|
|
rewrite *true_imp
|
|
end
|
|
end
|
|
|
|
end function
|