I tried looking it up online but I have found nothing, therefore I am unsure whether what I am doing is right, I'd like to also know what I am doing wrong in the proof.
Remember that if two permutations are conjugate, they have the same parity. Consider the partition induced on $S_n$ by the conjugacy relation $S_n/C_{S_n}$. We know that the number of even permutations is exactly half the order of the $S_n$, hence if we consider $Q = \{[f_i]_{C_{S_n}}|f_i \text{ is even}\}$ we get that $|\bigcup Q|= |S_n \setminus (S_n \setminus A_n)| = \frac{|S_n|}{2} = |A_n|$. Consider now the partition $A_n/C_{A_n}$, we have that $[f_i]_{C_{A_n}} \subseteq [f_i]_{C_{S_n}} \in Q$, if there were an $x\in [f_i]_{C_{S_n}} \setminus [f_i]_{C_{A_n}}$, then we would get $|\bigcup Q|= \sum_{i} |[f_i]_{C_{S_n}}| > \sum_{i} |[f_i]_{C_{A_n}}| = |\bigcup A_n/C_{S_n}| = |A_n|$, hence a contradiction. Therefore it must be $[f_i]_{C_{A_n}} = [f_i]_{C_{S_n}}$.
I'm not convinced of the proof, yet I am not sure what I might be doing wrong. Thanks.