Suggest Vaccination To Treat Dog Bite

My son was bitten by our house dog (vaccinated) the dog was provoked while sleeping. He tore his face near the eye. It s about an inch long and a flap about 1cm open. Skin is still attached. He washed washed with antibacterial soap and water for about 3 min and added a generous amount of Neosporin. His last tetanus shot was approximately 3 years ago. My son is 20 and refuses to go to the hospital.Is what he did enough?