Theorem: A function is invertible if and only if it is bijective
I understand this theorem, but one thing I feel is missing is that all elements in the domain have to map to some element in the co-domain. I do not feel like this theorem captures that. Am I right to say that all elements of the domain have to map to an element in the co-domain? If so, how does the theorem enforce this? It seems to me that the theorem enforces all the elements in the co-domain to be mapped to by some element in the domain - a unique one in particular - but not that all elements of the domain have to map to an element in the co-domain.
By default (in mathematics) all functions are total functions unless specifically identified as partial functions. This means a function is a relation for every element of its domain.
In some areas of computability theory, functions are partial by default. For instance, when discussing semidecidability, the subject matter is partial functions.