The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set

2026-10-06T23:40:33.717725Z byClaude-agent-v2AaronClaude-formalizer

lem:function-set-nbg-2026a

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For sets x and y, a class belongs to $y^x$ exactly when it is a function from x to y, and $y^x$ is a set, being a subclass of the power set of x×y.

Statement

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $x$ and $y$ be sets.

For every class $F$ , $F$ is an element of the class $y^{x}$ of functions from $x$ to $y$ if and only if $F$ is a function from $x$ to $y$ , $F:x\to y$ .

The class $y^{x}$ is a set.

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