Understanding Proof by Contradiction in Advanced Calculus

[Calculuser]

I was studying the first chapter "Sets and Structures" of the "A Course in Advanced Calculus - Robert S. Borden". I faced a difficulty at the part of the proof of contradiction.

I got confused at what this B= \{x \in A : x \not\in f(x) \} is and
how it's true that If~y \in A ~\text{is such that}~f(y)=B, \text{where is y? It must be either in}~B~\text{or in} A \setminus B.
Can anyone explain what's going on here?

 

[gopher_p]

Since $f:A\rightarrow\mathcal{P}(A)$, $f(x)$ is a subset of $A$ for each $x$. So for each $x\in A$, one of $x\in f(x)$ or $x\not\in f(x)$ is true. $B$ is the subset of $A$ where $x\not\in f(x)$ is true.

Since $B$ is a subset of $A$ - and thus $B\in\mathcal{P}(A)$ - it is reasonable to ask if $B$ is in the range of $f$. It's going to turn out that it's not, and that is the essence of the remainder of the proof and the "source" of the contradiction.

Assuming that $B$ is in the range of $f$, there is $y\in A$ such that $f(y)=B$. Forget about what $f$ and $y$ and $B$ are for a moment; that's just the definition of range from pre-calc. Then either $y\in f(y)=B$ or $y\not\in f(y)=B$ (note that $y\not\in B\Rightarrow y\in A \setminus B$); remember, since $f(y)=B$ is a subset of $A$ and $y\in A$, one of those has to be true. But either way you go, you end up with an absurdity; either $$y\in f(y)\Rightarrow_1 y\in B\Rightarrow_2 y\not\in f(y)$$ or $$y\not\in f(y)\Rightarrow_2 y\in B\Rightarrow_1 y\in f(y)$$ where the $\Rightarrow_1$ implications are "true" by virtue of the fact that $f(y)=B$ and the $\Rightarrow_2$ implications are true from the definition of $B=\{x\in A:x\not\in f(x)\}$.