I have no idea what "goes away in a bubble" means. If the air is not vented then it will form a bubble at the top of the tank but that is not "going away"!
The pressure of the air in the tank will be given by the "ideal gas law", PV= nRT. As the tank is filled with water, the V, the volume of the air in the tank, will be the initial volume of the tank minus the new volume of water. "n" is the number of molecules of air in the tank, "R" is the "gas constant", and "T" is the temperature of air in the tank which you can take to be constant here so "nRT" is a single constant, K. You have PV= K so that P= K/V. When P is air pressure, V is the volume of the tank. You can calculate K from that. As V decreases P must increase by P= K/V.
If the tank is vented so that, by "bubble away", you mean that the air bubbles up out of the tank to the top of the water, then the pressure of the water in the tank will just be the pressure of the water at the bottom of the dive.
