Suppose $A$ is a subset of the Cartesian product $X\times Y$, and $f$ is a function from $Y$ to some set. Is it acceptable to write $\{f(y):(x,y)\in A\}$ instead of $\{f(y):\exists x\ (x,y)\in A\}$? The logic behind the second notation seems to be to first form the set $\{y:\exists x\ (x,y)\in A\}$, and then replace each $y$ by $f(y)$. Then the first notation is justified by the following: replace each pair $(x,y)\in A$ by its second coordinate, and then replace that again by its image under $f$.
Motivation: in the theory of iterated forcing, a $\mathbb{P}*\dot{\mathbb{Q}}$-generic filter $K$ over the ground model $M$ induces a $\mathbb{P}$-generic filter $G$ over $M$ and a $\dot{\mathbb{Q}}_G$-generic filter $H$ over $M[G]$. Most sources seem to write $H:=\{\dot{q}_G:\exists p\ (p,\dot{q})\in K\}$.