It means "Is" in English :)

온통 세상이 그대인데? above sentence means :"A whole world doesn't change anything." 그대만 보이는데? above sentence means :"I can just see you." Korean language sometimes has and use a lot of meaning. Above cases, they mean " the present situation" as same "~이다"

Will you eat? I already ate. 먹을래? 나 이미 먹었는데? Will you see the movie ‘The ghost’? I already saw it. ‘유령’이라는 영화 볼래? 나 이미 봤는데?