Heat Lamp and Fan in Bathroom Help

I want to put a heat lamp and fan combo into my bathroom basement. To save cutting more holes in my house would I be able to tie the vent duct into the dryer vent. The dryer duct vents outside through the joist that I want to put my fan.

Not a good idea. The duct would likely be undersized, when the dryer was running, it could flow into the bathroom as the backflow dampers on those things isn't the greatest. A dryer doesn't normally have a backflow preventer (damper), so the bathroom would try to vent through the dryer as well.