Lazav the multifarius Copy persistence

if [[Lazav the multifarius]] copy a [[Dimir infiltrator]] and then it get destroyed, do he remain a copy of [[Dimir Infiltrator]] in the command zone\graveyard.

lets say i copy [[Mirror-mad Phantasm]] and decide to put lazav in the deck, it he still a copy of Mirror mad phantasm in the deck?

Thank you for your time.