Axiom Schema of Replacement

The axiom schema of replacement states that if F is a definable class function, as above, and A is any set, then the image F is also a set. This can be seen as a principle of smallness: the axiom states that if A is small enough to be a set, then F is also small enough to be a set. It is implied by the stronger axiom of limitation of size.

Because it is impossible to quantify over definable functions in first-order logic, one instance of the schema is included for each formula φ in the language of set theory with free variables among w1, ..., wn, A, x, y; but B is not free in φ. In the formal language of set theory, the axiom schema is:

\begin{align} \forall w_1,\ldots,w_n \, \forall A \, ( \\ &\Rightarrow \exist B \, \forall y \, ) \end{align}

Read more about Axiom Schema Of Replacement:  Axiom Schema of Collection, Example Applications, History and Philosophy, Relation To The Axiom Schema of Specification

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