15 lines
604 B
Text
15 lines
604 B
Text
import data.set
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open set function eq.ops
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variables {X Y Z : Type}
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lemma image_compose (f : Y → X) (g : X → Y) (a : set X) : (f ∘ g) '[a] = f '[g '[a]] :=
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setext (take z,
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iff.intro
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(assume Hz : z ∈ (f ∘ g) '[a],
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obtain x (Hx₁ : x ∈ a) (Hx₂ : f (g x) = z), from Hz,
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have Hgx : g x ∈ g '[a], from in_image Hx₁ rfl,
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show z ∈ f '[g '[a]], from in_image Hgx Hx₂)
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(assume Hz : z ∈ f '[g '[a]],
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obtain y [x (Hz₁ : x ∈ a) (Hz₂ : g x = y)] (Hy₂ : f y = z), from Hz,
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show z ∈ (f ∘ g) '[a], from in_image Hz₁ (Hz₂⁻¹ ▸ Hy₂)))
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