So I've been doing some exercises in Marker's Book, and I came across this particular exercise on expanding a saturated model $M$ of cardinality $\kappa$.
Let $L^*$ expand $L$ and $M$ a saturated $L-$structure. The author enumerates $(\phi_\alpha : \alpha < \kappa)$ of $L^*_{M}$- sentences, and gives a sketch of the proof in form of exercises. But I'm having some trouble understanding his hint for part b.), namely
"Show that if $\phi_{\alpha}$ is $\exists v\ \psi(v)$" and $T_\alpha + \{\phi_\alpha\} + T + Diag_{el}(M)$ is satisfiable, then for some $a\in M$, $T_\alpha + \{\phi_\alpha, \psi(a)\} + T + Diag_{el}(M)$, is also satisfiable."
For those who might not have the book, $T$ here refers to a $L^*$-theory, and $T_\alpha$ is some theory that is consistent with $T + Diag_{el}(M)$. Now in his hint, the author says:
"Let $A\subset M$ be the parameters from $M$ occuring in $T_\alpha + \{\phi_\alpha\}$ , and let $\Gamma(v)$ be all the $L_A$-consequences of $T_\alpha + \{\phi_\alpha, \psi(v)\} + T + Diag_{el}(M)$. Show that $\Gamma(v)$ is satisfiable and hence, by saturation, must be realized by some $a \in M$"
Edit: I realized I made a mistake in my proof in satisfiability of $\Gamma(v)$ Hence I will rephrase my question explicitly as follows:
1.) How do I proceed to show that $\Gamma(v)$ is satisfiable 2.) Why is it that if $\Gamma(v)$ is satisfiable, then by saturation we get that it must be realized by some $a\in M$.
Any help or insight to shed some light is appreciated.
Cheers
The $\mathcal{L}^*_M$-theory $T_\alpha\cup \{\phi_\alpha\}\cup T\cup \text{Diag}_{\text{el}}(\mathcal{M})$ is satisfiable, so it has a model $\mathcal{N}$. Since $\mathcal{N}|_{\mathcal{L}}\models \text{Diag}_{\text{el}}(\mathcal{M})$, we may assume $\mathcal{M}\preceq \mathcal{N}|_{\mathcal{L}}$. And since $\mathcal{N}\models \phi_\alpha$, $\mathcal{N}\models \psi(b)$ for some $b\in N$.
Let $A\subseteq M$ be the set of parameters from $\mathcal{M}$ occurring in formulas in $T_\alpha\cup \{\phi_\alpha\}$. Since $|T_\alpha| < \kappa$, $|A|<\kappa$. Let $p(x) = \text{tp}_{\mathcal{L}}(b/A)$. Since $\mathcal{M}$ is $\kappa$-saturated, $p(x)$ is realized in $\mathcal{M}$ by some $a\in M$.
It remains to show that the $\mathcal{L}^*_M$-theory $T_\alpha\cup \{\phi_\alpha,\psi(a)\}\cup T\cup \text{Diag}_{\text{el}}(\mathcal{M})$ is satisfiable. By compactness, it suffices to show that for any formula $\chi(a,\overline{c})\in \text{Diag}_{\text{el}}(\mathcal{M})$, where $\chi$ is an $\mathcal{L}_A$-formula and $\overline{c}\in M\setminus A$, the partial $\mathcal{L}^*_A$-type $T_\alpha\cup \{\phi_\alpha,\psi(x)\}\cup T\cup \{\chi(x,\overline{z})\}$ is satisfiable.
We have $\mathcal{N}\models T_\alpha\cup \{\phi_\alpha\}\cup T$. Interpreting $x$ as $b\in N$, $N\models \psi(b)$. And $\exists \overline{z}\, \chi(x,\overline{z})\in \text{tp}(a/A) = \text{tp}(b/A)$, so there is some tuple $\overline{c}'\in N$ such that $N\models \chi(b,\overline{c}')$. Interpreting $\overline{z}$ as $\overline{c}'$, we are done.
Note that I presented the argument here in a (superficially) different way than in Marker's hint. Rather than considering the set $\Gamma(v)$ from the hint, I took a witness $b$ in the elementary extension $\mathcal{N}$ and considered the complete type of $b$ over $A$. I think this approach makes the compactness argument at the end a bit clearer.