Question

There is a case brought in the first chapter of Arakhin that speaks about a woman who was in labor and died on the Sabbath: they bring a knife, cut open her womb, and remove the fetus. From here they bring proof that one may desecrate the Sabbath for a fetus. The Ran raises the argument that once the woman has died, the fetus is no longer considered its mother's thigh, but rather truly like a newborn. My question is: what really changed in essence when the mother died? (The claim is that it became independent, but that does not really sound reasonable, since that capacity, if it existed, was there even when the mother was alive.)