Transitive models of ZFC with satisfaction of formulas without any models(?)

Question

So i have started chapter 2 of Jech's Set Theory on Transitive models of ZFC. There are many parts which i don't understand. First let me give some context:

We define restricted quantifiers to be all quantifiers of the form $\forall u \in X$ or $\exists u \in X$. And next we call a formula $\psi$ a restricted formula if it has no quantifiers other than restricted quantifiers.

Lemma: If $M$ is a transitive class and $\psi$ a restricted formula, then for all $x_1, ... , x_n \in M$,
$M \models \psi(x_1, ... , x_n)$ iff $\psi(x_1, ... , x_n)$

According to the lemma every transitive model satisfies the axiom of extensionality. Because: $$ [(\forall u \in X)(u \in Y) \wedge (\forall u \in Y)(u \in X)] \rightarrow X=Y$$ is a restricted formula.


My questions:
1. How is $\psi$ valuated in the lemma at the right side of the equivalence at the absence of a model? Is it valuated under the model that we assume exists for the consistency of ZFC?

2. The axiom of extensionality in the second part is missing the $\forall X \forall Y$ part. Even if we forget about that, since $X$ and $Y$ are free variables of $\psi$ then when we want to check satisfaction we must add the respective quantifiers. So the axiom of extensionality isn't a restricted formula. Am i wrong? Or am i missing something?

Thanks for your patience.

Answer 1

1. Yes, we assume to work within a model of ZFC.
2. For the extensionality, $X$ and $Y$ are considered as free variables, and we can evaluate them as arbitrary elements of $M$.