does a faster lens mean better autofocusing in low light ?

still considering a m4/3 (maybe OMD?) and much of my shooting is street ... and those streets can be dark... and shooting later in the day.

thefast lenses are way more expensive than the slower ones (duh, ya i know) but since i dont usually carry a tripod, a faster lens would help give a higher shutter speed... one good reason.

however,

as my title asks, i dont expect an OMD to focus as well in low light as my D7000, so if i dont want a great disappointment, would a faster lens help in the focus department ?

from my internet travels, i seem to have read that the more light that reaches the sensor, the easier the focusing... does that hold true for these little ILC cameras ? i suppose the physics should be about the same ?... or are the focus mechanisms just not up to par with d7000s etc.. and faster lenses may a little better but not for difference in price..

most people buy fast lenses for DOF, i guess ?

(finally figured out this lens equivalent stuff. i do most of my shooting at 18-35, which is 27mm-50mm so on like an OMD, i'd need around a 12mm - 25mm). i think panasonic has faster lenses than olympus)