= Solution
With the same parameter $\alpha=(\omega_1)^M$, let
$$
\varphi_2(\alpha)\equiv\text{“there is no function }f\text{ with }\operatorname{dom}f=\omega\text{ and }\operatorname{ran}f=\alpha\text{.”}
$$
This formula is <downward absolute formula>[downward absolute]: if the larger transitive model has no such function, then neither can the smaller model, since any witness in the smaller model would remain a witness in the larger one. The model $M$ satisfies $\varphi_2(\alpha)$, while the generic surjection $\bigcup G$ makes it false in $M[G]$. Therefore it is not upward absolute.
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