Definition:Diffeomorphism on Image

Open Sets in $\R^n$
Let $n$ and $k$ be natural numbers.

Let $U \subset \R^n$ be an open set.

Let $f : U \to \R^n$ be a mapping.

Then $f$ is a $C^k$-diffeomorphism on its image $f(U)$ is open and $f : U \to f(U)$ is a $C^k$-diffeomorphism.