According to Wikipedia ("Image (mathematics)", 17-Mar-2024, https://en.wikipedia.org/wiki/ImageSupport_(mathematics)): "... evaluating a given function at each element of a given subset of its domain produces a set, called the "image of under (or through) ". Similarly, the inverse image (or preimage) of a given subset of the codomain of is the set of all elements of the domain that map to the members of ." The preimage of a set under a function is often denoted as "f^-1 (B)", but in set.mm, the idiom is used. As a special case, the idiom for the preimage of a function value at under a function is (according to Wikipedia, the preimage of a singleton is also called a "fiber").
We use the label fragment "preima" (as in mptpreima) for theorems about preimages (sometimes, also "imacnv" is used as in fvimacnvi), and "preimafv" (as in preimafvn0) for theorems about preimages of a function value.
In this section, will be the set of all preimages of function values of a function , that means is a preimage of a function value (see, for example, elsetpreimafv): .
With the help of such a set, it is shown that every function can be decomposed into a surjective and an injective function (see fundcmpsurinj) by constructing a surjective function and an injective function so that ( see fundcmpsurinjpreimafv). See also Wikipedia ("Surjective function", 17-Mar-2024, https://en.wikipedia.org/wiki/Surjective_function (section "Composition and decomposition"). This is different from the decomposition of into the surjective function (with for ) and the injective function , ( see fundcmpsurinjimaid), see also Wikipedia ("Bijection, injection and surjection", 17-Mar-2024, https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection (section "Properties").
Finally, it is shown that every function can be decomposed into a surjective, a bijective and an injective function (see fundcmpsurbijinj), by showing that there is a bijection between the set of all preimages of values of a function and the range of the function (see imasetpreimafvbij). From this, both variants of decompositions of a function into a surjective and an injective function can be derived:
Let be a decomposition of a function into a surjective, a bijective and an injective function, then with (an injective function) is a decomposition into a surjective and an injective function corresponding to fundcmpsurinj, and with (a surjective function) is a decomposition into a surjective and an injective function corresponding to fundcmpsurinjimaid.