-
Notifications
You must be signed in to change notification settings - Fork 0
/
Imagen_de_la_imagen_inversa.thy
63 lines (52 loc) · 1.41 KB
/
Imagen_de_la_imagen_inversa.thy
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
(* Imagen_de_la_imagen_inversa.thy
Imagen de la imagen inversa
José A. Alonso Jiménez
Sevilla, 10 de junio de 2021
---------------------------------------------------------------------
*)
text \<open>------------------------------------------------------------------
Demostrar que
f ` (f -` u) \<subseteq> u
---------------------------------------------------------------------\<close>
theory Imagen_de_la_imagen_inversa
imports Main
begin
section \<open>1\<ordfeminine> demostración\<close>
lemma "f ` (f -` u) \<subseteq> u"
proof (rule subsetI)
fix y
assume "y \<in> f ` (f -` u)"
then show "y \<in> u"
proof (rule imageE)
fix x
assume "y = f x"
assume "x \<in> f -` u"
then have "f x \<in> u"
by (rule vimageD)
with \<open>y = f x\<close> show "y \<in> u"
by (rule ssubst)
qed
qed
section \<open>2\<ordfeminine> demostración\<close>
lemma "f ` (f -` u) \<subseteq> u"
proof
fix y
assume "y \<in> f ` (f -` u)"
then show "y \<in> u"
proof
fix x
assume "y = f x"
assume "x \<in> f -` u"
then have "f x \<in> u"
by simp
with \<open>y = f x\<close> show "y \<in> u"
by simp
qed
qed
section \<open>3\<ordfeminine> demostración\<close>
lemma "f ` (f -` u) \<subseteq> u"
by (simp only: image_vimage_subset)
section \<open>4\<ordfeminine> demostración\<close>
lemma "f ` (f -` u) \<subseteq> u"
by auto
end