Why does set-builder notation have this particular property?

Question

Consider $\{z\in \mathbb{Z} : z = ax + by\;\exists\, x, y \in \mathbb{Z}\}$ and $\{ax + by : \forall\, x, y \in \mathbb{Z}\}$. These two sets are one and the same. But noticed that the quantifiers flipped. Why does this happen?

Answer 1

Let a,b be integers.
The correct set construction is
{ ax + by : x,y in Z }.

Answer 2

Neither of your proposals is correct. The first one, however, comes close to an expression which is valid: $$\{z\in\mathbb{Z}: \exists x,y\in\mathbb{Z}(z=ax+by)\}.$$ However, you'll commonly - more commonly perhaps - find expressions like $$\{ax+by: x,y\in\mathbb{Z}\}.$$ These expressions are a bit syntactically weird: there's an implicit variable binding going on after the colon. Intuitively, when we introduce some new variables to the left of the colon and then include constraints on those variables to the right of the colon, those variables range over all objects satisfying those constraints.

A bit more snappily: given a term $t(x_1,...,x_n)$, the expression $$\{t(x_1,...,x_n): [stuff]\}$$ is accepted notation for the set $$\{y: \exists x_1,...,x_n([stuff]\mbox{ and }y=t(x_1,...,x_n)\}.$$