How can someone be alive if they're brain dead?

The answer to this very complicated question is both yes and no. My father was only 56 years old and had won the battle over stage 4 colon cancer when he went to the doctors office for a simple procedure he had had done several times before. This time was different - he went into cardiac arrest and his heart wasn't able to be started for over 20 min. He was pronounced brain dead. His body otherwise was healthy including his heart. He was brain dead because his heart had quit beating for 20 minutes and therefore his brain went without oxygen for 20 minutes thus causing him to become brain dead. He was on a ventilator for 48 hours and the doctor said even if he were to wake up right then he would not know anything or anyone. So basically be a vegetable. I know my father would not want to live this way. The doctors gave us a choice and we chose to take him off the ventilator and he passed away peacefully in about 5 minutes. Until then, his body was alive but his brain was dead.