Fill in some of 3.5, fix numbering - #268
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gaearon
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Aug 4, 2025
| z ∈ (SetTheory.Set.slice x Y) ↔ ∃ y:Y, z = (⟨x, y⟩:OrderedPair) := replacement_axiom _ _ | ||
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| /-- Definition 3.5.2 (Cartesian product) -/ | ||
| /-- Definition 3.5.4 (Cartesian product) -/ |
| apply ext; | ||
| aesop | ||
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| /-- Example 3.5.5 / Exercise 3.6.5. There is a bijection between `X ×ˢ Y` and `Y ×ˢ X`. -/ |
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Calling out the future exercise explicitly so it's clear why we're not filling this in.
| invFun := sorry | ||
| left_inv := sorry | ||
| right_inv := sorry | ||
| /-- Example 3.5.5. A function of two variables can be thought of as a function of a pair. -/ |
gaearon
commented
Aug 4, 2025
| tuple a = tuple b ↔ a = b := by sorry | ||
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| /-- Example 3.5.8. There is a bijection between `(X ×ˢ Y) ×ˢ Z` and `X ×ˢ (Y ×ˢ Z)`. -/ | ||
| noncomputable abbrev SetTheory.Set.prod_associator (X Y Z:Set) : (X ×ˢ Y) ×ˢ Z ≃ X ×ˢ (Y ×ˢ Z) where |
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I've fully filled this in because it wasn't a "why" and it wasn't part of an exercise.
gaearon
commented
Aug 4, 2025
| noncomputable abbrev SetTheory.Set.prod_equiv_prod (X Y:Set) : | ||
| ((X ×ˢ Y):_root_.Set Object) ≃ (X:_root_.Set Object) ×ˢ (Y:_root_.Set Object) where | ||
| toFun := fun z ↦ ⟨(fst z, snd z), by simp⟩ | ||
| invFun := fun z ↦ mk_cartesian ⟨z.val.1, z.prop.1⟩ ⟨z.val.2, z.prop.2⟩ |
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Filled these in because it's auxiliary to the book.
gaearon
commented
Aug 4, 2025
| right_inv := sorry | ||
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| /-- Example 3.5.5. A function of two variables can be thought of as a function of a pair. -/ | ||
| noncomputable abbrev SetTheory.Set.curry_equiv {X Y Z:Set} : (X → Y → Z) ≃ (X ×ˢ Y → Z) where |
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Filled this in because it isn't a "why" or an exercise.
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There's a few changes in here:
mem_insertlemma so we can unwrap >3-item sets without mentioning the instance.curry,uncurry,curry_uncurry, anduncurry_curryinto an equivalence. This matches how other equivalences are shown and is IMO a bit more intuitive. It took me a while to see why they're defined at all.