Look at my post 7
Under IPC you can use combination drain and vent method on the island sink. The pipe must be 2 inch and no garbage disposals installed. Vent location doesn't matter.
I would look and make sure your plumber has pitch on the pipe 1/4" per foot. But overall I don't see an issue.
I'm just trying to explain it to Chandler He lives in an IPC state.