Let $\phi: \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a diffeomorphism and $M\subset \mathbb{R}^n$
M is a k-manifold if and only if $\phi(M)$ is a k-manifold. Prove it.
So what I did was try to prove it using the following property of manifolds:
there exists a local diffeomorphism $f$ near $x_0$ such that $x \in M$ if and only if $f(x)\in \mathbb{R}^k \times ${0}$^{n-k}$ for all x near $x_0$ for every $x_0 \in M$
so I used that and obtained that I need to prove that there exists a local diffeomorphism $h$ so that $x \in \phi(M)$ if and only if $h(x)\in \mathbb{R}^k \times ${0}$^{n-k}$ for all x near $x_0$ for every $x_0 \in \phi(M)$
so I took $h=f\circ \phi$, is that enough?
To prove the other direction we just take $h^{-1}$ because since $h=f\circ \phi$ is a diffeomorphism , $h^{-1}$ is also a diffeomorphism.
Is that right?