The 513 stands for 5 pickups 13 sounds.
You have the normal 5 way like on your Ibanez and the other switch is a 3 way, choosing from single coil/vintage output humbuckers/modern (hot) output humbuckers
The way it is on the pic it is bridge humbucker at full output.
More info: http://www.prsguitars.com/513/
That's a great and versatile setup. First I was doing the math and I thought 3x5=15, but then I realized the middle position in the five way is just the middle single coil so it's the same single coil tone whether the other switch makes it single coil, regular hum, and hot hum.
I read that on top of that the single coils don't hum so either they are stacked or maybe there's side by side coils under the cover. The whole thing is ingenious and far better than the 5 way varitone from Gibson which puzzled many a player.
When you are in a noisy gig and amps and/or pedals are not up to par, or you have to suddenly lengthen the set due to being enlisted for more time, the last thing you want is a confusing guitar. I like "single coil" which I would use to cut through too much mud, regular humbucker to get a medium baseline tone to start with, and hot for if and when you need it. Different rooms and audiences live can dictate that and it's hard to predict. I have seen where the hottest dual EMG 81 ESP just fell flat because it was mud to begin with and the tone needed would be more akin to a regular three single coil strat. That would have cut through. Sometimes the boring middle single coil and nothing else gives enough fatness for a rich tone yet cuts through and is mostly left in on a mixed three pickup setup.
When I play in my job at a rest home to do music therapy, there are tons of noises and interruptions and what gets through the mix is single coils. It's not unlike, well, drunk and disorderly people. What a lot of people who play in different live situations will find is that a standard three humbucker Les Paul or SG guitar has a blind spot for cutting through if audience is not quiet and still. Not every PA has enough power to overwhelm that in a way it doesn't turn everything into mud. There's are reason that a good PA operator gets paid more than the hired musicians at times, but that's another thread! Those people are masters.
Back to single coils and cutting through a non-quiet audience, what I did for one of my dual humbucking guitars was put in a push-pull on the volume to allow for three single coil tones to add to the standard three humbucker tones. I simplified the wiring and bypassed the tone control and went straight to three way switch and to output jack. This works for live situations quite well but wouldn't be the best for studio. For what I do practicing, jamming, and live, it's a perfect setup. But your guitar seems to anticipate both live and keeping single coil in mix on all three settings as life jacket, and the studio where you can access the widest range of tones yet avoid any 60 cycle hum on any of the 13 settings. I don't know why Fender, Gibson, or Ibanez haven't figured out this dual switch system for pickups.
Lastly, my guess for what PRS did was have the middle single coil operate as a stack if used alone in position 3 of 5 but somehow be cut to a single coil and by RWRP with one of the coils of the humbucker when put in position 2 or 4 of the five way switch. That way there's no way for hum to cut through. Of course you could also operate that middle single coil as a stack all the time but it may not be as glassy when used with one of the two full sized humbuckers. But PRS may have some sort of filter in there to give it the sound of a true single coil even if it is a stacked single coil. Anyway this type of wiring workaround is fascinating to me and a hum-less HSH setup is a great idea whose time has come.