You didn't say "reduces the crime rate outside of prison." I assumed that's what you meant, but it's not clear that ignoring the crime rate inside prison is a reasonable statistic.
People in prison also, presumably, eventually get out, and a claim that prison officials can accurately deduce the likelihood of recidivism, and whether it has been decreased rather than increased by time in prison, is far from clear.
Finally, putting lots of people in prison has an effect on people outside of prison. For example, it is possible—though, again, I don't know; citations are needed—that high incarceration rates lead to more crime outside, since, if a member of a community has a good chance of going to prison whether or not they commit a crime, then prison can cease to have a meaningful deterrent effect in that community.
That's the sort of statement that seems plausible, and even intuitive, but probably needs a citation. It wouldn't wholly surprise me if it were true, though at moral and economic cost; but it would surprise me even less if it were false.