Is a pseudorandom function (PRF) also a one-way function (OWF)? If yes, how can we proof that a PRF $f_k$ is a OWF? If no, what is the closest work?

Let $f_k$ be a PRF. We claim that $f_k$ is a OWF. PROOF let $f_k$ is not a OWF, there exists a $PPT$ algorithm $A$ that can invert $f_k$ with non-negligible advantage. Even if we know the input $x$ for given $f_k(x)$ with a non-negligible advantage, how can we claim that we can distinguish $f_k(x)$ from random with non-negligible advantage? Here, a key $k$ is still secret.

A PRF is a keyed function, or alternatively it is a family of functions indexed by a key. A OWF is a single public function. There is a mismatch between these two things.

AES is a secure PRF (function family). Does that mean $\textsf{AES}(k,\cdot)$ is a OWF? The adversary attacking a OWF is allowed to depend on the function it is attacking. So the OWF adversary can know $k$ in the context of the OWF security game. Of course if you know $k$ you can easily invert AES -- it is not a OWF, not for any fixed $k$.

I would say that $\textsf{AES}(k,\cdot)$ is hard to invert for a random and secret choice of $k$, if you have only black-box access to the function. But not a OWF according to the standard OWF security definition.