Equality of morphism of representations of $S_n$ and $A_n$

Question

Let $S_n$ be the symmetric group and $A_n$ the Alternating group.

Let $(V,\rho)$ be a simple representation of $S_n$.

Let $(W,\pi)$ be a simple representation of $A_n$.

Suppose $V=W$. I saw in some proof that for $x\in A_n: \rho(x) = \pi(x)$.

Is that correct? I can't see why they must be equal for $x\in A_n$

Thank you for your help.