I'm trying to understand Scott's proof of the incompatibility of axiom of constructibility and the existence of a measurable cardinal. I'm stuck in the use of Łoś's Theorem in the universe. Jech's construction go as follow:
Let $S$ be a set and let $F$ be the class of all functions with domain $S$. In $F$, consider the equivalence relation
$$f =^* g \Leftrightarrow \{x \in S : f(x) = g(x) \} \in U$$
And then Jech says that the class of a function under that relation is
$$[f] = \{g : f=^*g \text{ and }\forall h(h=^* f \rightarrow \mathrm{rank} \ g \leq \mathrm{rank} \ h)\}$$
Now, why do we have to use $\forall h(h=^* f \rightarrow \mathrm{rank} \ g \leq \mathrm{rank} \ h)$?
You want to collect all the equivalence classes, but if each is a proper class you can't do that. Proper classes are not elements of other classes. On the other hand, "chopping" each equivalence class to a set allows you to proceed.
This is known as Scott's Trick. It is also useful for defining the cardinals in models where the axiom of choice fails.