Is this reasonable for £580?

Nah. It doesn cost that. And i wouldn't go for AMD cpu's. Atleast for current ones. I'm certain you can get better videocard(atleast someting around 7850 2gb) and better cpu for such price. All this cost somewhere around £330 combined.

At a quick glimpse it seems ok, I would personally get a better GPU if you are investing so much in a new computer, also I would suggest doubling that ram. Im assuming as you are going 6 cores you are looking at mutlitasking.

Yeah looks fine to me.
You can do without the aftermarket Heatsink if you're not overclocking.
You also didn't mention a power supply unit.

I'd change the fx6350 for Phenom II x4 970 and pick HD7790 or hd7850 instead of 7770. You'll need that graphics power to run games in 1080p, and Phenoms are comparable to FXes (in gaming) and a lot cheaper.

Is that a pre-built pc, built by your choice of parts or parts bought independantly?
If it is a pre-built pc it will cost more than built by parts.
It is to expensive for the components you are getting.
I bought a Phenom II x4 965 BE 3.4Ghz, 8gb ddr3 1333Mhz, HD 7770, Motherboard and a OCZ 500w PSU for about 300€ (255£).
My 965BE is about the same as your FX 6350, and between 1333 and 1600Mhz you will not notice the diference.
My advice is to talk to a frinds who understands computer building better than you to help you get the right parts for your needs and the best performance/price computer possible. To buy my new parts i spent about a month researching and investigating.
Good Luck!

You can do better, I spec'd this up for a friend. Roughly the same price but more powerful.

switch the cpu for an AMD Phenom II X4 955 Black Edition and use the savings for an AMD Radeon HD 7850

CPU: Intel Core i5 3470, Motherboard: ASRock Z75 Pro3, RAM: Patriot Signature 2 x 4GB DDR3-1600, Hard Drive: Western Digital Caviar Blue 1TB 3.5" 7200RPM, Video Card: GeForce GTX 660 2GB, Case NZXT Tempest 210 ATX Mid Tower, Power Supply: XFX 550W 80 PLUS Bronze Certified ATX12V / EPS12V, Optical Drive: Lite-On iHAS124-04 DVD/CD Writer, Operating System: Microsoft Windows 7 Home Premium SP1 OEM 64-bit, Total 780 USD or 587 EURO

http://pcpartpicker.com/p/Z87S

For ~£50 less, this one I knocked together gives you a better processor, a better graphics card and a bigger hard-drive (2TB). Everything else I left as similar as I could / put an alternative in.

If you went with the same processor as you've got in the main post, and a smaller HDD than I've put, you could bring the price down by ~£40.

For a measly £20 more, you could get yet more graphics power...

For the full £575 you can get a even better processor and a GTX 660. Or go with the 7870 for £578.

Note, all 3 builds with the Radeon HD 7850, 7870 in come with a free copy of Far Cry 3, Bioshock Inifinte and Tomb Raider AMD Never Settle

yes it is