EDB — 1WS

[1WS]Let $f:\mathbb{N}\to \mathbb{N}$ be an assigned function and $I$ its image, prove that $A\subseteq \mathbb{N}$ exists such that $f{|}_A$ is injective and $f(A)=I$. (Hint it may be useful to know that the usual order of $\mathbb{N}$ is a well-order cf [07R]↺↻ and [26Y]↺↻).

Note: The result is true for any function $f:A\to B$, but the proof requires the axiom of choice.