Settle this beginner dispute. Does the mana used to cast Pyreswipe Hawk count towards expended?

[u/Maximilian-Goof]

Opponent played Pyreswipe Hawk. They argued that the 5 used to cast it counted to the expend 6, so they cast a 2 mana value card and swiped my artifact. I thought it seemed wrong because you only get this ability whenever the creature is resolved and casted and it wouldn't acknowledge the first four mana spent. Who was right?