∄ x ∈ ℕ: x² = 5
Unless you consider the domain of f the "renamed" range, in which case yes tautologically it is bijective.
Unless you consider the domain of f the "renamed" range, in which case yes tautologically it is bijective.
You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. Any surjective function can be made non-surjective by considering a larger range.
It's worse than just irrelevant. The claim is that we have two sets, ℕ and S, that are the same size. The argument that they are the same size is the observation that we've demonstrated a bijection between them.
No one ever mentioned a function from ℕ to ℕ, because that would tell you nothing about S. It would be like saying "in order to show that 5 is odd, observe that 2 and 2 are both even". The idea is gibberish.