The surjectivity of a Lie group homomorphism $\phi:G\to H$ with $H$ connected.

Question

Let $G,H$ be Lie groups with $H$ connected and assume $\phi:G\to H$ is a Lie group homomorphism which is a diffeomorphism onto an open neighbourhood $\mathcal{V}\subset H$ around $e_H$. Does it follow that $\phi$ is surjective?

Answer 1

Yes, since the image contains an open neighbourhood of the identity, and such a neighbourhood generates the group.

a neighbourhood of identity $U$ generates $G$ where $G$ is a connected lie group