Note first that $U$ and $U^{\prime}$ are normal by [EGA, IV.6.5.4]. Since direct images commute with flat base change we have $f_{*}\mathcal{O}_{U^{\prime}}=\mathcal{O}_{U}$, which implies that $f$ has connected fibers as a consequence of the theorem on formal functions (cf. [EGA, III.4.3.2]). Since $U$ is connected and $f$ is closed, surjective and has connected fibers, it follows that $U^{\prime}$ is connected, hence irreducible since it is normal. ∎