Mathematical Proof and the Principles of Mathematics/Sets/Replacement

Replacement

The Axiom Schema of Replacement says that if one replaces each of the elements of a set according to some formula, then the result is also a set.

Axiom Schema (Replacement)

Let $P(x,y,A)$ be a property such that for each $x\in A$ there is a unique $y$ such that $P(x,y,A)$ holds. There exists a set $B$ consisting of all the $y$ for which there exists some $x\in A$ such that $P(x,y,A)$ holds.

Technically the formula is allowed to have finitely many free variables, and is often written $P(x,y,w_{1},w_{2},\ldots ,w_{n},A)$.

As for the Axiom Schema of Comprehension, there is an axiom in the schema for every possible property $P(x,y,w_{1},w_{2},\ldots ,w_{n},A)$.

As for the Axiom of Foundation, most of mathematics can be accomplished without the Axiom Schema of Replacement. However, the axiom allows for the construction of certain infinite sets that are important in set theory itself.