How does bleed air maintain pressure for pressurization after being cooled? I found on aerosavvy.com/aircraft-pressurization the paragraph below, which talks about bleed air on airliners:


  Most modern airliners use bleed air from the compressor section of the engines to pressurize the cabin. This very hot air must be cooled to a comfortable temperature before it’s directed into the cabin.


If it is cooled too much it wouldn't be under high pressure, right?

The cabin needs high pressure air because the air at 30,000ft is at too low of a pressure, so they need to simulate a 6000-8000 feet pressure. This bleed air is very hot high pressure air, I understand you have to cool it because it is way too hot for anything to operate with it.

But if you cool it, doesn't it lose pressure, so it would be low pressure air which is not what we want? I don't know where I am getting confused at.
That's one of the reasons why bleed air is compressed again before cooling.


  In Air Cycle Machines, high-pressure bleed air from the engines is first passed through a compressor, further squeezing the already hot gas. It is then routed through a heat exchanger or two to remove heat. The now cooler but still highly compressed air then passes through an expansion chamber into a larger chamber. The combined effects of driving the turbine and expanding into a larger chamber dramatically cools the air (usually down close freezing; water traps are critical in the system to prevent freeze-up). (Source)


Simple diagram:



(Source)

The outflow valve is what regulates the pressure, it makes sure the right amount of air leaves the cabin.


  The automatic controller normally maintains the proper cabin pressure altitude by constantly adjusting the outflow valve position so that the cabin altitude is as low as practical without exceeding the maximum pressure differential limit on the fuselage.