For the sake of brevity let me use the following terms. A subset $X$ of $\bar{\mathbb{Q}}$ will be called "small" if for any number field $K$, the intersection $X\cap K$ is finite. Similarly, a set $Y\subset \bar{\mathbb{Q}}$ will be called "large" if for any number field $K$, $Y$ contains all but finitely many elements of $K.$ My question is whether the following claim is true.

Claim: Let $f\in \bar{\mathbb{Q}}[x, y]$ be a polynomial which is non constant in $y,$ and let $X\subset \bar{\mathbb{Q}}$ be a "small" set. Then the set $Y=\lbrace y\in \bar{\mathbb{Q}}$: there exists $x\in X$ such that $f(x,y)=0\rbrace$ is not "large".