Add utility sink to existing laundry plumbing

Hello all from Fort Worth Texas. New to this group and glad to have found it. I would like to add a utility sink to our existing laundry room and would like to request your expert advice in reference to connecting to the existing plumbing. The pictures you see are from when the house was being built and looking from the hallway into the laundry room (walls are covered now) one picture is looking from the laundry room entrance. We would have to pull some drywall from the bottom of the laundry wall in order to access the plumbing. There are original pictures and an edited picture with my proposed additions/changes. Can I please have you guys review them and advice if my proposed changes would work and would be code compliant? Thank you in advance.

Ed

That will work or you could add tee above the trap on the standpipe and pipe over from there if you’re close enough there would be no need to add the vent