Diffeomorphism preserves compact support of functions?

Question

Let $M$ and $N$ be two Riemannian manifolds which are diffeomorphic via a $C^k$ map $F:M \to N$.

Let $\phi \in C^0_c(M)$ be a continuous function with compact support in $M$. Is it true that its pushforward to $N$, $$\phi \circ F^{-1}$$ has compact support in $N$? How to prove it? If not what do I need to impose to make it true?

Answer 1

Hint By construction, the support of $\phi \circ F^{-1}$ is $F(\text{supp }\phi)$.