Definition 1.5.17.  Let X, Y be sets, f : X → Y be a function. We define:
f is surjective  :⇔  ∀ y ∈ Y : ∃ x ∈ X : f(x) = y  ⇔  Y ⊆ f(X)  ⇔  f(X) = Y
Equivalence.  No proof.
References.