For mixtures of Bose and Fermi alkali-metal atoms, the fermion degrees of freedom can be integrated out in their finite temperature partition function. The final result can be expanded as a power series in the boson density. Under appropriate conditions, the pairwise interaction between the bosons can be changed from positive to negative by the fermion-mediated term at a low enough temperature. The boson cloud may then collapse. For attractive fermion-boson interactions, there is also a collapse of the fermion cloud. This transition is first order (discontinuous) because the leading-order nonlinear interaction term between the bosons is third order in boson density with a negative coefficient. We discuss the finite-temperature phase diagram of this transition. Our result may provide for an explanation of recent experimental observations by Modugno and co-workers.