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Union_con_la_imagen.thy
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Union_con_la_imagen.thy
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(* Union_con_la_imagen.thy
-- Unión con la imagen
-- José A. Alonso Jiménez <https://jaalonso.github.io>
-- Sevilla, 22-abril-2024
-- ------------------------------------------------------------------ *)
(* ---------------------------------------------------------------------
-- Demostrar que
-- f ` (s \<union> f -` v) \<subseteq> f ` s \<union> v
-- ------------------------------------------------------------------ *)
theory Union_con_la_imagen
imports Main
begin
(* 1\<ordfeminine> demostración *)
lemma "f ` (s \<union> f -` v) \<subseteq> f ` s \<union> v"
proof (rule subsetI)
fix y
assume "y \<in> f ` (s \<union> f -` v)"
then show "y \<in> f ` s \<union> v"
proof (rule imageE)
fix x
assume "y = f x"
assume "x \<in> s \<union> f -` v"
then show "y \<in> f ` s \<union> v"
proof (rule UnE)
assume "x \<in> s"
then have "f x \<in> f ` s"
by (rule imageI)
with \<open>y = f x\<close> have "y \<in> f ` s"
by (rule ssubst)
then show "y \<in> f ` s \<union> v"
by (rule UnI1)
next
assume "x \<in> f -` v"
then have "f x \<in> v"
by (rule vimageD)
with \<open>y = f x\<close> have "y \<in> v"
by (rule ssubst)
then show "y \<in> f ` s \<union> v"
by (rule UnI2)
qed
qed
qed
(* 2\<ordfeminine> demostración *)
lemma "f ` (s \<union> f -` v) \<subseteq> f ` s \<union> v"
proof
fix y
assume "y \<in> f ` (s \<union> f -` v)"
then show "y \<in> f ` s \<union> v"
proof
fix x
assume "y = f x"
assume "x \<in> s \<union> f -` v"
then show "y \<in> f ` s \<union> v"
proof
assume "x \<in> s"
then have "f x \<in> f ` s" by simp
with \<open>y = f x\<close> have "y \<in> f ` s" by simp
then show "y \<in> f ` s \<union> v" by simp
next
assume "x \<in> f -` v"
then have "f x \<in> v" by simp
with \<open>y = f x\<close> have "y \<in> v" by simp
then show "y \<in> f ` s \<union> v" by simp
qed
qed
qed
(* 3\<ordfeminine> demostración *)
lemma "f ` (s \<union> f -` v) \<subseteq> f ` s \<union> v"
proof
fix y
assume "y \<in> f ` (s \<union> f -` v)"
then show "y \<in> f ` s \<union> v"
proof
fix x
assume "y = f x"
assume "x \<in> s \<union> f -` v"
then show "y \<in> f ` s \<union> v"
proof
assume "x \<in> s"
then show "y \<in> f ` s \<union> v" by (simp add: \<open>y = f x\<close>)
next
assume "x \<in> f -` v"
then show "y \<in> f ` s \<union> v" by (simp add: \<open>y = f x\<close>)
qed
qed
qed
(* 4\<ordfeminine> demostración *)
lemma "f ` (s \<union> f -` v) \<subseteq> f ` s \<union> v"
proof
fix y
assume "y \<in> f ` (s \<union> f -` v)"
then show "y \<in> f ` s \<union> v"
proof
fix x
assume "y = f x"
assume "x \<in> s \<union> f -` v"
then show "y \<in> f ` s \<union> v" using \<open>y = f x\<close> by blast
qed
qed
(* 5\<ordfeminine> demostración *)
lemma "f ` (s \<union> f -` u) \<subseteq> f ` s \<union> u"
by auto
end