Generally the power supplies that come with low cost cases should not be trusted. I noticed in the pics that that PSU claims to be able to put out 22 amps. If it can handle at 4870, it's gonna be close. That said, if it were my rig, I'd either look at a different card, or upgrade the power supply, but I would not run that card with the power supply in question.
But if you try to run it, and the power supply goes out on you, those cheaper ones are more dangerous because it could fry your entire rig if it goes. But you should be reasonably safe running a low power card like a 4670 on it.
not sure how that would ruin the whole computer. Wouldnt it need some short of surge of power to ruin the motherboard and rig? Isnt that what a fuse is for so the surge of power never makes it? guess im not under standing how the charge would jump?
Yup, I've actually had a case I think where I had a cheapo power supply possibly take out a mobo back in the day. Had it running for about a month, all of a sudden the mobo stopped recognizing IDE ports.