Need help adding Sink to laundry room

I'm trying to add a sink in my laundry room, one problem I'm having is how to vent the drain as there are water supply pipes (going to the upstairs) in the way preventing me from going straight across to tie into the venting for the washer and floor drain. So I was wondering if I can tee off the existing vent line running horizontally out of the room and send a pipe towards the ceiling to meet the vent pipe from my new sink. I would just build a soffit to cover it. See picture of drawing for better understanding.

Additionally to drain the sink can I tie into the horizontal part of the floor drain vent or do I tie into the 3" main drain line in the house. See picture this is the plumbing under laundry room in crawl space

Thank You for your help

 

The simplest way to do it would be to offset the waterlines to the back of the wall and run the vent to the opposite side where they pass.. Theres just enough room in a 3.5" stud bay for both to cross.

Or can you use an Air Admittance Valve / AAV / Studor vent? Not sure which plumbing code you fall under.

 

Thanks that does make sense aav would not be allowed here so offsetting them would be best. As far as the drain line where would you tie into ?