TheoremBase

Images of Sets under Functions Are Sets, and a Function with a Set Domain Is a Set

In NBG, replacement makes the image of a set under a function a set, and a function whose domain is a set is itself a set, being a subclass of the Cartesian product of its domain with its image.

Statement

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let FF be a function.

For every set xx, the image F[x]F[x] is a set.

If the domain dom⁡F\operatorname{dom}F is a set, then FF is a set.

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