Given a set function $f : X \to X$ let $\sim$ be the equivalence relation $x \sim f(x)$. Contextually, I am working with the coequalizer of $f$ and $1_X$. I want to have as much information about the set $X/\hspace{-3pt}\sim$ as possible.
So far it's only obvious to me that the classes $[x]$ contain the fibers of any representative and also contain the elements $f^n(y)$ for each $y \in [x]$ and for all $n \in \mathbb{N}$. This suggested to me that $$ [x] = \lbrace x, fx,f^2x, \ldots \rbrace \cup \left( \bigcup_{n=1}^{\infty} (f^{-1})^n \lbrace x \rbrace \right) $$
The inclusion of the set on the right in the set on the left is quick to see, but I can't seem to prove the other inclusion because it starts from a relation and tries to deduce an equality.
I tried a few examples like $f:\mathbb{N} \to \mathbb{N}$ with $f(n)=2n$, and some small finite sets and they fit into the picture above, but that was only because I had an explicit description of $f$. I'm sure there must be a counterexample to the equality above, but I can't come up with one.
Thanks.