This is what's holding me up.
I'm imagining a pure solvent in a flask, with a vacuum above it. There is enough space above the liquid for... let's say... a trillion gaseous solvent molecules. So in pure solvent, there will be a trillion molecules that escape the liquid and enter the gas phase. Once they reach that trillion number, the gas and liquid enter equilibrium. Those trillion gas molecules exert a certain pressure on the container - its vapor pressure.
Now let's say it is a solution instead. There is still space above the solution for a trillion solvent molecules - so the solvent molecules still can escape the solvent (even though it is at a slower rate because there are fewer spaces on the surface area for the solvent to escape). The solvent molecules still escape the liquid, albeit at a lower rate, until about a trillion gaseous molecules are above the liquid. At that trillion number, it would establish equilibrium again. It seems that if there is space for a trillion gaseous molecules, then there should be a trillion gaseous molecules.