HD 5770. Enough power supply?

you should be fine, the ATI 5770 is very power efficient

those are the ATI recommendations and totally useless. They are listed to cover crap PSU people buy for $20. the antec truepower series are very good PSU. your antec has 62a on the 12v rail, more then enough for the ATI 5770, in fact more then enough for four of them.
a system with with the ATI 5770 needs a decent 400w unit with 22a on the 12v. the following system below won't even break 300w with the ATI 5770 during a stress test

Core i7 Test System Specs
- Intel Core i7 965 Extreme Edition (Overclocked @ 3.70GHz)
- x3 2GB G.Skill DDR3 PC3-12800 (CAS 9-9-9-24)
- Asus P6T Deluxe (Intel X58)

I can understand your confusion, what you really have is a single 12v rail that has been split 4x, not four 12v rails. hopefully this helps you out.

What is "multiple +12V rails", really?

In most cases, multiple +12V rails are actually just a single +12V source just split up into multiple +12V outputs each with a limited output capability.

There are a few units that actually have two +12V sources, but these are typically very high output power supplies. And in most cases these multiple +12V outputs are split up again to form a total of four, five or six +12V rails for even better safety. To be clear: These REAL multiple +12V rail units are very rare and are all 1000W+ units (Enermax Galaxy, Topower/Tagan "Dual Engine", Thermaltake Tough Power 1000W & 1200W, for example.)

In some cases, the two +12V rail outputs are actually combined to create one large +12V output (Ultra X3 1000W, PC Power & Cooling Turbo Cool 1000W, for example.)

http://www.jonnyguru.com/forums/showthread.php?t=3990