Allah was before there was anything else, and He will remain after everything is gone. There is no life before Him or without Him. When we call upon Allah al-Hayy, we are recognizing Him as the absolute source of life and thus the source of all perfect qualities.
Al-Qayyum is the One by whom all things are eternally managed aright. He is the source of everything needed by the worshiper, from the One he or she worships.