Quick question about Bitcoin mining

Ok, so let's say I built a super cheapo system with an Intel Celeron or something like that in there, GB of RAM, and ya, and then added in...let's say....a 6670. Obviously the 6670 will be bottlenecked and I won't be able to get all of the performance out of it for gaming (which I wouldn't be doing on this machine anyways).

 

Here's my question, even though the 6670 would be bottlenecked, when I start Bitcoin mining, will the miner utility that I'm using use the entire power of the 6670 even though it's bottlenecked, or it will it just use the stuff that's usable, and then the bottlenecked stuff doesn't get used at all?

No, that works fine. GPU miner will run the thing at 100% whatever your other hardware