Let $M$ be a saturated model of ZFC and let $\kappa$ be any cardinal in $M$. Now let $\begin{align*} p_\kappa(f) &= \{ “f \text{ 1-1 function”} \wedge \operatorname{dom}(f) \subseteq \omega \wedge \operatorname{ran}(f) \subseteq \kappa \} \\ &\qquad\cup \{n \in \operatorname{dom}(f) : n \in \omega \} \cup \{ \alpha \in \operatorname{ran}(f) : \alpha \in \kappa \}. \end{align*}$ $p_\kappa(f)$ is fin. realized in $M$ and therefore it is realized in $M$ (since $M$ is saturated and $\kappa < \operatorname{card}(M)$). So there exist $f_\kappa : \omega \to \kappa$ that is $1-1$ and onto, which of course is a contradiction. What am I doing wrong?
Note : This is question is related to another question that I asked earlier, "A question on saturated models of ZFC". I don't know if I should merge the two questions.