Not really, although it sounds like an odd arrangement. We don't know how clean and stable this 5V supply is, so first make sure it definitely doesn't go above 5V, whether or not the PSU is switched on. If you are satisfied it is safe and won't blow up the Arduino, then try it. As you say, you may need to add an additional resistor in parallel with the Arduino power rails in order to draw enough current to switch on the PSU.
Obviously it's "safer" to use a separate 5V supply for the Arduino (or a higher voltage one that you connect to Vin rather than 5V). But I understand the attraction of making use of what's already there.