Consider the following theorem of Erdos and Hajnal:
Definition. For any set $x$, a function $f$ is called ${\omega} $-Jonsson iff $f$: $^{\omega}x$ $\rightarrow$ x and whenever $y$$\subseteq$$x$ and |$y$|=|$x$|, $f^"$$^{\omega}y$=$x$ ($f^"$$^{\omega}x$=range ($f$$\upharpoonright$$^{\omega}x$)).
Theorem. For every infinite cardinal $\lambda$, there is an $\omega$-Jonnson function over $\lambda$.
The proof of this theorem uses the Axiom of Choice. In $ZFC$, can one provide a 'simple' proof of this theorem in which the Axiom of Choice (or a sentence equivalent to it) is an explicit step in the proof?