AT&T iPhone 5 activated to verizon?

I have a unlocked AT&T iPhone 5 but the carrier on the phone still says AT&T?

Can I just take it into Verizon and tell them it's unlocked and they can switch my iPhone 4 over to the iPhone 5?

If it is truly unlocked and the IMEI is clean, yes. New SIM card and let the programming begin. Make a backup of all of your iPhone 4 data on iTunes before going in. When you get home you can the sync the 5 to iTunes a restore from iTunes backup. It will then pump all the info over--just to make sure you don't lose anything. Best luck.