Reducing pressure in a binary liquid solution Consider the following diagram for a binary liquid system comprised of A and B, where A is the more volatile component.

My professor says

Consider an isopleth from $a$ to $a_4$. At the start, the solution is in pure liquid phase at state $a$.
As we reduce the pressure, the solution starts to boil at $a_1$. The composition of the liquid solution is denoted by the point $a_1$ which is the same as the overall composition of the solution since the solution has only just started to boil. The composition of the first trace of vapor formed is $a_1'$.
As the we reduce the pressure, the solution continues to boil to reach point $a_2''$. We may draw a tie line to find the composition of the liquid $a_2$ and the composition of the vapor $a_2'$. At every pressure where the solution exists as a mixture of liquid and vapor, we may use a tie line to find out its composition in the liquid and vapor phase.
Upon reaching $a_3'$, the solution has just been completely changed into vapor phase and the last trace of liquid has the composition $a_3$. After all the solution is vaporized at $a_4$, the composition of the vapor is again the overall composition of the solution.

I am slightly confused about we are able to make the pressure drop from $a_1$ to $a_3'$. My reasoning is: since the liquid and vapor are in equilibrium, shouldn't enough liquid keep on evaporating to maintain a constant vapor pressure?
Consider a system with a variable volume (e.g., a syringe, but on a larger scale). As the volume increases, liquid becomes vapor without changing the amount of material present. 

In a system where there is only one component, changing the volume of the container will change the fraction of material in the vapor vs the amount of material in the liquid. But as you concluded, the vapor pressure will remain constant (assume constant temperature, since it was not mentioned), because there is an equilibrium between liquid and vapor for the one material in each phase.

However, when there are two components of different volatility, the vapor composition varies, depending on how much vapor there is, compared to the fraction left as liquid. So the equilibrium is no longer between a vapor and a liquid of the same composition. 

It might be easier to visualize the change going on in the liquid/vapor if you change the dimension of the vertical axis to read 1/Volume. Then the vapor over the solution is (intuitively) richer in A, always (again, intuitively), until the liquid is all gone, when the vapor attains the original composition. Then, BTW, you realize that the pressure was changing while you changed the volume, and pressure might be a more relevant parameter.