Logical form of statement I'm reading the book How to Prove It and a question is given to write out the logical form of the below definition in set-theoretic notation.
Definition: $y \in \{\sqrt[3]{x} \mid x\in\mathbb{Q}\}$
Given solution: $(\exists x\in\mathbb{Q})(y=\sqrt[3]{x}).$
How is this solution derived? 
I'm not sure of the steps taken. I can "sort of" read both equations and have sense that they are both correct but I'm not sure how the logical conclusion is reached?
Update : 

 A: Let's discuss what these notations mean.  There are two (but only two!) important principles operating here.
Principle 1: When we write $$\{x \mid \text{$x$ is  ugly}\}$$ that means the set of all $x$ such that $x$ is ugly, or we could say for short the set of all ugly things.  This notation goes hand-in-hand with the $\in$ notation; they define each other, because  $$y\in \{x \mid \text{$x$ is ugly}\}$$ means exactly that $$y\text{ is ugly},$$ nothing more nor less. And this is true in general.  If $P(x)$ is any property that $x$ might or might not have, then $$\{x\mid P(x)\}$$ is the set of things with property $P$, and $$y\in \{x\mid P(x)\}$$ means exactly that $P(y)$ holds, nothing else.
Now let's consider the set that contains all the people who are mothers of ugly people.  Let's write $\def\M#1{\operatorname{Mother}(x)}\M x$ to mean the mother of $x$.  What does it mean to say that $m$ is the mother of an ugly person?  It means that there is some ugly person $x$ of whom $m$ is the mother, so there is some $x$ such that $x$ is ugly and $m = \M x$.  In symbols “there is some” is just $\exists$, so we can translate “$m$ is the mother of an ugly person” into symbols as:  $$(\exists x)( \text{$x$ is ugly and } m=\M x)$$ and we can write the set of all such people as $$\{ m \mid (\exists x)( \text{$x$ is ugly and } m=\M x) \}\tag{1}.$$
This is the set of all mothers of ugly people. This construction comes up so often that there is a common abbreviation for it:
$$\{\M x \mid \text{$x$ is ugly} \}\tag{2}$$
We read this as “the set of all mothers of ugly people”, and (Principle 2)
1 and 2 mean exactly the same thing.  Unlike principle 1, this really isn't anything fundamental, it's just a convention about what $\{f(x) \mid \cdots\}$ is supposed to mean.
Now suppose we say that $y$ is in this set; that is, that $y$ is in the set of mothers of ugly people:
$$y\in\{\M x \mid \text{$x$ is ugly} \}.$$
Because (2) is just an abbreviation for (1), this is the same as $$y\in\{m \mid (\exists x)( \text{$x$ is ugly and } m=\M x) \}$$ and because of the way $\in$ and $\{x\mid\cdots\}$ go together (principle 1), this means nothing more nor less than 
$$ (\exists x)( \text{$x$ is ugly and } y=\M x)$$ which says that there is some ugly person $x$ of whom $y$ is the mother.  This should make sense: If $y$ is a member of the set of the mothers of ugly people, then there is some ugly person $x$ of whom $y$ is the mother.
Your example is exactly like these in its form.  You have $$y \in \{\sqrt[3]{x} \mid x\in\mathbb{Q}\}$$ which reads as “$y$ is a member of the set of cube roots of rational numbers”.  This is an abbreviation for $$y\in \{c \mid (\exists x\in\Bbb Q)(c=\sqrt[3]x)\}$$ which is “$y$ is a member of the set of all $c$ for which there is some rational number $x$ of which $c$ is the cube root” just as  $(1)$ is an abbreviation of $(2)$.
But because of the way $\in$ and $\{c\mid\cdots\}$ go together (principle 1 again), this means exactly that $$(\exists x\in\Bbb Q)(y=\sqrt[3]x),$$
nothing more nor less.  This reads as “There is some rational number $x$ of which $y$ is the cube root”.
You said you can  "sort of" read both equations and have sense that they are both correct, and that is good, because it is very important—even more important than knowing how to push the symbols around.  But you also need to know how to push the symbols around.
A: Well, you have $y \in \{\sqrt[3]{x} : x\in\mathbb{Q}\}$. $\exists x\in\mathbb{Q},y=\sqrt[3]{x}$ basically states that, since $y$ is a member of the aforementioned set, $y$ is equivalent to some $x$ that prescribes to the definition in the set constructor.
In other words, if $\exists x\in\mathbb{Q}$, then the latter part of the definition is fulfilled. Then, if $y=\sqrt[3]{x}$, the first part of the definition is fulfilled. Since $y$ fulfills both conditions of the set constructor, it must be in the set described.
