The answer to your question is "yes, you can do that."
You would first make sure that the iPad Air has "enable remotes" turned on from the "Tools" -> Presentation Tools menu. Then the iPad mini (assuming its on the same network, and has Keynote open and the "remote" section active) will appear on the iPad Air as a option for a remote device. A confirmation code will appear and then all you have to do is open the presentation on the iPad Air, and the iPad mini will be able to control it (and you can also set this up to be done vice-versa, of course).