If that smaller vertical pipe is a vent, and the piece coming off of it is 2", the maximum distance you can be from that vent is 5' for a p-trap. But, if you run a revent vertical from the trap arm of the new sink, going up at least 42" above the floor (or 6" above the flood rim of the sink, whichever is higher), and run it back over to the vent line, then you can make that drain run as long as you want.
The only consideration I see is that that arm is already fairly high above the floor, and if you nee to run it 10' over to where you want the sink, it will go up another 2.5" minimum to maintain the 1/4" per foot slope required, and then it may be too high for your sink to install a trap. It looks like it's about the right height IF your sink was essentially installed about the end of the existing stub, and even then, it may be high if the sink is very deep. Some people like to use kitchen cabinets for the extra height in a bathroom (nice if you are tall), and that might give you a little more leeway over a 30-32" tall vanity.