i'm flying jet airways from the US to india, landing in Mumbai but final destination is hyderabad. I have a 9 hr layover in mumbai before the Hyd flight on Jet.

1. Once I arrive, can I check in the luggage to Jet domestic & get a boarding pass after I take the luggage from my international flight, pass thro the customs, etc.?
2. Once I check my luggage in, I'd like to exit the airport to go into the city, and come back in time for the domestic flight. is there an issue with exiting?

thanks.