It's explained in Velleman's how to prove book that $\exists x \in AP(x)$
means that there is at least one value of x in the set A such that P(x) is true.
Then they go and define this logical form in one of the following pages:
$\exists n \in N(x = n^2 )$
Instead of P(x) here where something is true or false, how can I think about that for an assignment statement $x = n^2$ ? For all the values of n in the set N, there will be some value of x anyway.
$x=n^2$ is not an assignment statement. It is an assertion that the expressions $x$ and $n^2$ have the same value. This is a statement that is either true or false, depending on the values of $x$ and $n^2$.
Mathematics does not have assignment statements. $=$ in mathematics is always a comparison, never an assignment. It is like the
==operator in the C, Python, or Java programming languages, not like the=operator in those languages.