In an academic research paper in computer science, I have a formula using the set builder notation in order to define a set $A$, as follows:
$$A = \{ y(b_i)=a_i | b_i \in B \}.$$
In the particular example, I want to convey in a concise way that each element $a_i \in A$ results from the transformation given by the function $y(b_i)$ for all $b_i \in B$.
Is that an appropriate and correct formulation, especially the part on the left-hand side (LHS) of the set builder?
That is, in set theory, is it allowed/correct to use an equality on the LHS of the set builder in the same way as above?
The notation you have suggested parses, and would likely be understood by most readers. However, a couple of things would make it more clear:
You are defining an object $a_i$, which is an element of $A$. It is easier to read from left-to-right, so I would suggest writing $a_i = y(b_i)$ (instead of $y(b_i) = a_i$).
The index $i$ is doing nothing for you in this notation. I would drop it, and write $a = y(b)$ with $b \in B$.
The spacing around the vertical bar is wrong. Either use
\mid, or\ |\(or just use a colon).My version of that notation would be
$$ A = \{ a = y(b) \mid b \in B \}. $$
You could also move the specification of $a$ to the right-hand side and write
$$ A = \{ a \mid a = y(b), b \in B\}, $$
though I think that my preference would be for the former.
As noted in the comments, there are other common notations which would give the same set. If $y$ is a function, then $A$ is the image of $B$ under $y$. A very short notation for this is $$ A = y(B). $$ You can also completely leave off the identity in the set-builder notation and write $A = \{ y(b) \mid b \in B\}$.
The choice of notation here is going to be largely up to your taste, and up to the taste of whatever reviewer is reading your work. And remember that the goal is clarity of exposition—good notation explains what you are doing in a clear and precise way; bad notation causes confusion. As long as you are not causing confusion, you're fine.