Can you Airplay from an iPhone to Apple Tv without your router being connected to the Internet?

I have an iPhone 4 and an old router laying around. Sadly, there is no actual internet connection in the house, but I want to be able to stream from my iPhone to an Apple TV. Can I just plug in the router and use (I think its called the LAN?) that the router puts out to connect the two devices over Airplay? I wouldnt need an Internet connection for Netflix or anything, I just want to mirrior my iPhone onto my TV.

Thanks,

McFierce