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Interseccion_con_la_imagen.thy
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Interseccion_con_la_imagen.thy
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(* Interseccion_con_la_imagen.thy
-- Intersección con la imagen
-- José A. Alonso Jiménez
-- Sevilla, 19 de junio de 2021
-- ------------------------------------------------------------------ *)
(* ---------------------------------------------------------------------
-- Demostrar que
-- (f ` s) \<inter> v = f ` (s \<inter> f -` v)
-- ------------------------------------------------------------------ *)
theory Interseccion_con_la_imagen
imports Main
begin
(* 1\<ordfeminine> demostración *)
lemma "(f ` s) \<inter> v = f ` (s \<inter> f -` v)"
proof (rule equalityI)
show "(f ` s) \<inter> v \<subseteq> f ` (s \<inter> f -` v)"
proof (rule subsetI)
fix y
assume "y \<in> (f ` s) \<inter> v"
then show "y \<in> f ` (s \<inter> f -` v)"
proof (rule IntE)
assume "y \<in> v"
assume "y \<in> f ` s"
then show "y \<in> f ` (s \<inter> f -` v)"
proof (rule imageE)
fix x
assume "x \<in> s"
assume "y = f x"
then have "f x \<in> v"
using \<open>y \<in> v\<close> by (rule subst)
then have "x \<in> f -` v"
by (rule vimageI2)
with \<open>x \<in> s\<close> have "x \<in> s \<inter> f -` v"
by (rule IntI)
then have "f x \<in> f ` (s \<inter> f -` v)"
by (rule imageI)
with \<open>y = f x\<close> show "y \<in> f ` (s \<inter> f -` v)"
by (rule ssubst)
qed
qed
qed
next
show "f ` (s \<inter> f -` v) \<subseteq> (f ` s) \<inter> v"
proof (rule subsetI)
fix y
assume "y \<in> f ` (s \<inter> f -` v)"
then show "y \<in> (f ` s) \<inter> v"
proof (rule imageE)
fix x
assume "y = f x"
assume hx : "x \<in> s \<inter> f -` v"
have "y \<in> f ` s"
proof -
have "x \<in> s"
using hx by (rule IntD1)
then have "f x \<in> f ` s"
by (rule imageI)
with \<open>y = f x\<close> show "y \<in> f ` s"
by (rule ssubst)
qed
moreover
have "y \<in> v"
proof -
have "x \<in> f -` v"
using hx by (rule IntD2)
then have "f x \<in> v"
by (rule vimageD)
with \<open>y = f x\<close> show "y \<in> v"
by (rule ssubst)
qed
ultimately show "y \<in> (f ` s) \<inter> v"
by (rule IntI)
qed
qed
qed
(* 2\<ordfeminine> demostración *)
lemma "(f ` s) \<inter> v = f ` (s \<inter> f -` v)"
proof
show "(f ` s) \<inter> v \<subseteq> f ` (s \<inter> f -` v)"
proof
fix y
assume "y \<in> (f ` s) \<inter> v"
then show "y \<in> f ` (s \<inter> f -` v)"
proof
assume "y \<in> v"
assume "y \<in> f ` s"
then show "y \<in> f ` (s \<inter> f -` v)"
proof
fix x
assume "x \<in> s"
assume "y = f x"
then have "f x \<in> v" using \<open>y \<in> v\<close> by simp
then have "x \<in> f -` v" by simp
with \<open>x \<in> s\<close> have "x \<in> s \<inter> f -` v" by simp
then have "f x \<in> f ` (s \<inter> f -` v)" by simp
with \<open>y = f x\<close> show "y \<in> f ` (s \<inter> f -` v)" by simp
qed
qed
qed
next
show "f ` (s \<inter> f -` v) \<subseteq> (f ` s) \<inter> v"
proof
fix y
assume "y \<in> f ` (s \<inter> f -` v)"
then show "y \<in> (f ` s) \<inter> v"
proof
fix x
assume "y = f x"
assume hx : "x \<in> s \<inter> f -` v"
have "y \<in> f ` s"
proof -
have "x \<in> s" using hx by simp
then have "f x \<in> f ` s" by simp
with \<open>y = f x\<close> show "y \<in> f ` s" by simp
qed
moreover
have "y \<in> v"
proof -
have "x \<in> f -` v" using hx by simp
then have "f x \<in> v" by simp
with \<open>y = f x\<close> show "y \<in> v" by simp
qed
ultimately show "y \<in> (f ` s) \<inter> v" by simp
qed
qed
qed
(* 2\<ordfeminine> demostración *)
lemma "(f ` s) \<inter> v = f ` (s \<inter> f -` v)"
proof
show "(f ` s) \<inter> v \<subseteq> f ` (s \<inter> f -` v)"
proof
fix y
assume "y \<in> (f ` s) \<inter> v"
then show "y \<in> f ` (s \<inter> f -` v)"
proof
assume "y \<in> v"
assume "y \<in> f ` s"
then show "y \<in> f ` (s \<inter> f -` v)"
proof
fix x
assume "x \<in> s"
assume "y = f x"
then show "y \<in> f ` (s \<inter> f -` v)"
using \<open>x \<in> s\<close> \<open>y \<in> v\<close> by simp
qed
qed
qed
next
show "f ` (s \<inter> f -` v) \<subseteq> (f ` s) \<inter> v"
proof
fix y
assume "y \<in> f ` (s \<inter> f -` v)"
then show "y \<in> (f ` s) \<inter> v"
proof
fix x
assume "y = f x"
assume hx : "x \<in> s \<inter> f -` v"
then have "y \<in> f ` s" using \<open>y = f x\<close> by simp
moreover
have "y \<in> v" using hx \<open>y = f x\<close> by simp
ultimately show "y \<in> (f ` s) \<inter> v" by simp
qed
qed
qed
(* 4\<ordfeminine> demostración *)
lemma "(f ` s) \<inter> v = f ` (s \<inter> f -` v)"
by auto
end