Does using the fan above the stove reduce heat in kitchen from the oven & save on AC cost in a small apartment?

My husband just came into the kitchen yelling at me because I was using the ventilator fan above the stove as I was roasting a turkey in the oven. He states that I am wasting electricity. I tried explaining to him that the heat from the oven rises and goes out with the fan and that helps save the AC costs because the heat doesn't go out into our small one bedroom apartment as much and therefore trigger the AC to come on more often. He says I am wrong. Who's right ?