Extension of a bounded and uniformly continuous function

Extension of a bounded and uniformly continuous function

I read in a book the following assertion:
Let $\Omega\subset\mathbb{R^n}$ be a open set, then any bounded and
uniformly continuous function in $\Omega$ has a unique bounded continuous
extension to $\overline{\Omega}$.
I can't see this readly, first I thought was a simple consequence of the
Tietze extension theorem, but I didn't have any progress. Someone could
help me?
Thanks!