Boolean-Valued models on a proper class universe

Question

In Jech's Set Theory he defines a Boolean-Valued model to consist of a Boolean universe $A$ and two functions of two variables with values a complete Boolean algebra $B$. The functions are $||x = y||$ and $||x\in y||$. However, when $A$ is a proper class, we cannot define these two functions as sets. Thus when defining the boolean valued model $V^B$, we are actually recursively defining $b = ||x\in y||$ as a formula $\phi(b,x,y)$. But then Jech defines $$||\exists x\psi(x,a_1,\dots,a_n)|| = \sum_{a\in A}||\psi(a,a_1,\dots,a_n)||.$$ By Tarski's undefinability of truth, we shouldn't be able to define this on a proper class $A$. I think that this is because we cannot find a set-like well founded relation to justify the recursion. This leads me to believe that when defining a Boolean-Valued model, the universe $A$ must be a set. What is actually going on here? My guess is one way to get around this is to define $V^B$ inside $M$. That way outside of $M$ we know that our universe $M^B$ is a set. He does this in later chapters but never explicitly says this. Is that what is actually happening?