# Is the domain of an one-to-one function a set if the target is a set?

This is probably very naive but suppose I have an injective map from a class into a set, may I conclude that the domain of the map is a set as well?

-

2
Perhaps you should clarify what is meant by "map" in this question. The answers using replacement seem to be assuming that the map is defined by a formula in the language of set theory. – ccc Feb 23 '12 at 15:19

## 3 Answers

If a function $f:A\to B$ is injective one, we can assume without loss of generality that $f$ is surjective too (by passing to a subclass of $B$), therefore $f^{-1}:B\to A$ is also a bijection.

If $B$ is a set then every subclass of $B$ is a set, so $f^{-1}:B\to A$ is a bijection from a set, and by the axiom of replacement $A$ is a set.

-

I say you can, since an injective map $f:A\to B$ is isomorphic to both its image (which is a set) and its domain.

-

Define $g$ on the range of $f$ such that $g(f(x)) = x$. This is well-defined because $f$ is injective. The domain of $g$ is equal to the range of $f$, which is a set.

Therefore by the axiom of replacement, or maybe the axiom of union applied to $\bigcup_{y\in{\rm Range}f} \{g(y)\}$, the range of $g$ is a set. But the range of $g$ is precisely the domain of $f$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 19, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9717344045639038, "perplexity_flag": "head"}