modular arithmetic - Basic question about mod

I'm having a tough time understanding why $(a^x \bmod p)^y \bmod p$ is equal to $a^{xy}\bmod p$.
Does this have a mathematical proof? Please advise.

Answer

Since this was migrated from SO, I suspect that by "$n\bmod m$", you mean the remainder when $n$ is divided by $m$, typically with $0\le n