if $f$ is holomorphic in $\Bbb D$ then it is equal to its power series in all $\Bbb D$?

Question

I saw a solution to some question somewhere and this was implicitly used.

I know holomorphic functions are given LOCALLY by their taylor series, but from the fact that $f\in Hol(\Bbb D)$ can we conclude that $f$ is given in all $\Bbb D$ by its taylor series at $0$?

In particular, how can we be sure that its taylor series at $0$ converges in all $\Bbb D$ and only in some disk around $0$ $B\subsetneq \Bbb D$?