Simon Cowell, Lauren Silverman and baby Eric
02/17/2014 at 06:00 PM EST
Baby made three for Simon Cowell and girlfriend Lauren Silverman on Valentine's Day, and the new family is already on the move.
Two days after welcoming Eric Philip into the world, the music mogul was seen Sunday leaving the hospital with mom and baby.
"Simon was carrying a sleeping Eric in his car seat," an onlooker tells PEOPLE, adding, "Lauren looked absolutely amazing and very pretty – it was hard to believe she had given birth less than 48 hours prior. She kept smiling."
A beaming new dad, Cowell, 54, posted photos
of his son on Twitter the same day, writing, "I never knew how much love and pride I would feel."
News first broke in July that Cowell and Silverman were expecting a baby.
A source tells PEOPLE that Eric has already changed the couple for the better.
"It seems their son has already brought them closer," says the source. "He just really had no idea that he would feel so emotionally attached and protective of his son."
Given his busy work schedule (he's in the midst of filming the U.K. hit Britain's Got Talent
), the source says Cowell will frequently be hopping the pond to spend time with his new son.
"He has no choice but to continue with auditions and then filming in England," adds the source. "But it will be very difficult for him to leave his son so he plans to fly back to New York whenever he can for the next few weeks, even if he can only spend a few hours with his son. And when Eric is bigger, Lauren will fly to England."