37 lines
1.4 KiB
Text
37 lines
1.4 KiB
Text
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import data.set.basic
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open set
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definition preimage {X Y : Type} (f : X → Y) (b : set Y) : set X := λ x, f x ∈ b
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example {X Y : Type} {b : set Y} (f : X → Y) (x : X) (H : x ∈ preimage f b) : f x ∈ b :=
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H
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theorem preimage_subset {X Y : Type} {a b : set Y} (f : X → Y) (H : a ⊆ b) : preimage f a ⊆ preimage f b :=
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λ (x : X) (H' : x ∈ preimage f a), show x ∈ preimage f b,
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from @H (f x) H'
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example {X Y : Type} {a b : set Y} (f : X → Y) (H : a ⊆ b) : preimage f a ⊆ preimage f b :=
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λ (x : X) (H' : x ∈ preimage f a),
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have f x ∈ a, from H',
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have f x ∈ b, from mem_of_subset_of_mem H this,
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this
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example {X Y : Type} {a b : set Y} (f : X → Y) (H : a ⊆ b) : preimage f a ⊆ preimage f b :=
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λ (x : X) (H' : x ∈ preimage f a),
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have f x ∈ b, from mem_of_subset_of_mem H H',
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this
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example {X Y : Type} {a b : set Y} (f : X → Y) (H : a ⊆ b) : preimage f a ⊆ preimage f b :=
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λ (x : X) (H' : x ∈ preimage f a),
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@H (f x) H'
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lemma mem_preimage_of_mem {X Y : Type} {f : X → Y} {s : set Y} {x : X} : f x ∈ s → x ∈ preimage f s :=
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assume H, H
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lemma mem_of_mem_preimage {X Y : Type} {f : X → Y} {s : set Y} {x : X} : x ∈ preimage f s → f x ∈ s :=
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assume H, H
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example {X Y : Type} {a b : set Y} (f : X → Y) (H : a ⊆ b) : preimage f a ⊆ preimage f b :=
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take x, assume H',
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mem_preimage_of_mem (mem_of_subset_of_mem H (mem_of_mem_preimage H'))
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