Lemma $14.18$ in Jech's Set Theory states the following when expositing forcing.
If $W$ is a set of pair wise disjoint elements of a Boolean algebra $B$ and if $a_u$, $u\in W$, are elements of $V^B$, then there exists some $a\in V^B$ such that $u\leq ||a=a_u||$.
I have a question about the first line of the proof.
Proof: Let $D = \bigcup_{u\in W}\text{dom}(a_u)$, and for every $t\in D$, let $a(t) = \sum\{u\cdot a_u(t):u\in W\}$.
It seems like $a(t)$ isn't necessarily defined. For example there may be some $t\in \text{dom}(a_u)$ and $t\notin \text{dom}(a_v)$ for $u,v\in W.$ Thus $a_v(t)$ is undefined. What is happening here?