Hi, Tom.
Although the engine bleed air is cooled somewhat before it reaches the aircon "pack", a lot more processing is carried out. On a 747-400, the air is pre-cooled with the pack heat exchanger, then compressed, then run through the heat exhanger again, then fed through a turbine and then into the cabin (in the latter stages, expanding it greatly). This cooling, compression, cooling and expansion allows the very hot bleed air to provide very cold air to the cabin. Even on 100 degree F days, you may see/feel minute chunks of ice being spat out the airconditioning ducts. The process is not unlike your home refrigerator (which compresses, cools and expands a refrigerant gas/liquid). Other than atmospheric air, however, no gas is necessary in an airplane pack.
On the ground, the cooling airflow over the heat exhanger may be assisted with a fan (In the air, of course, the airflow over the heat exchanger will be more than adequate without the fan...especially at Mach .83 at 39,000'!).
The aircon packs may be shut down during certain phases of your trip (e.g. engine start and takeoff), but hopefully, the cool air in the cabin will not disappear straight away.
Rgds.
Q.