Suppose $M \models T$, $A \subseteq M$, and $f:A \to M$ is partial elementary. Let $b\in M-A$. Let
$$
\Gamma= \big\{\phi(v,f( \bar{a} ): \bar{a} \in A^m \wedge M\models\phi(b, \bar{a} )\big\}
$$
If $\phi(v,f( \bar{a} ) )\in \Gamma $, then $M\models\exists v \phi(v, \bar{a} ) $ and hence, because $f$ is partial elementary, $M\models\exists v \phi(v, f(\bar{a} ) ) $. Thus, because $\Gamma$ is closed under conjunctions, $\Gamma$ is satisfiable.
Could you please explain why $\Gamma$ is satisfiable!