Contemporary models of bounded plasmas assume that the target plasma electron temperature far exceeds the temperature of the cold electrons emitted from the target, T_{emit}. We show that when the sheath facing a collisional plasma becomes inverted, the target plasma electron temperature has to equal T_{emit} even if the upstream plasma is hotter by orders of magnitude. This extreme cooling effect can alter the plasma properties and the heat transmission to thermionically emitting surfaces in many applications. It also opens a possibility of using thermionic divertor plates to induce detachment in tokamaks.