Replacement in the well-founded part
= Replacement in the well-founded part
Without assuming <foundation>, the <class in set theory> formed by the <cumulative hierarchy> satisfies <replacement>. Relativize a functional <logical formula> to that definable <class in set theory> and use ambient <replacement> to collect its range. Every output belongs to the hierarchy <class in set theory>, and the rank-bounding result places the range <set> itself in that <class in set theory>.