surjective_of_nonempty_fiber_of_isConnected

English

If X has a nonempty fiber and A is connected, then F.map f is surjective for f: X ⟶ A.

Русский

Если фибра X непустая и A связный, то F.map f сюръективен для f: X ⟶ A.

LaTeX

$$$\operatorname{Surjective}\bigl(F.map f\bigr)$$$

Lean4

theorem surjective_of_nonempty_fiber_of_isConnected {X A : C} [Nonempty (F.obj X)] [IsConnected A] (f : X ⟶ A) :
    Function.Surjective (F.map f) :=
  by
  have : Epi f := epi_of_nonempty_of_isConnected F f
  exact surjective_on_fiber_of_epi F f

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