How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set? Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity 

There exists an inductive set. 

with the Axiom of Dedekind-infinite set

There exists a set equipollent with its proper subset.


How can I define the set of natural numbers $\mathbb{N}$ in this setting, and prove that it is the unique minimal inductive set?

The answers given above mention natural numbers, but there is no need: read Dedekind's "Wass sind und wass sollen die Zahlen?". In there he does what CopyPasteIt describes and very carefully proves that all desirable properties of $\mathbb{N}$ hold in the smallest set $N$  that contains $a_0$ and is closed under $f$. Unlike Andres he refrains from writing $N=\{f^n(a_0):n$ is a natural number$\}$ because there is no need (and it would be self-referential as $n=f^n(a_0)$ in hindsight).
The map $g$ can be defined by recursion on $N$ by $g(a_0)=\emptyset$ and $g(f(x))=g(x)\cup\{g(x)\}$; here Replacement comes into play to show that $g$ and hence its range is a set.
Or you can deduce from Dedekind's work that $N$ is an infinite well-ordered set and observe that its order-type is an inductive set.
