card_fiber_eq_of_iso — Mathlib

English

The cardinality of the fiber is preserved under isomorphisms.

Русский

Кардинальность фибра сохраняется при изоморфизмах.

LaTeX

$$$\\forall X,Y\\; (i:X\\cong Y) \\Rightarrow |F(X)| = |F(Y)|$$$

Lean4

/-- The cardinality of the fiber is preserved under isomorphisms. -/
theorem card_fiber_eq_of_iso {X Y : C} (i : X ≅ Y) : Nat.card (F.obj X) = Nat.card (F.obj Y) :=
  by
  have e : F.obj X ≃ F.obj Y := Iso.toEquiv (mapIso (F ⋙ FintypeCat.incl) i)
  exact Nat.card_eq_of_bijective e (Equiv.bijective e)

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