In page 15 of "Lie groups beyond an introduction" by Prof. Knapp, Corollary 0.20 states that the exponential map (of linear Lie algebra $\mathfrak g$) generates the identity component $G_0$ of a closed linear group $G$. The proof goes like
1 by continuity of the exponential map $exp(\mathfrak g)$ is connected
2 $exp(\mathfrak g)$ contains a neighborhood of 1 in the group (local diffeomorphism)
3 the smallest subgroup containing a nonempty open set in $G_0$ must be $G_0$
3 doesn't look obvious to me though, I don't know how to prove that $exp(\mathfrak g)$ is a subgroup. In addition, if we can show it's open, then it's an open closed connected subgroup, so it must be $G_0$ since $G_0$ is connected.