To elaborate this we need the concept of a bijection.

Recall that to prove this we need to exhibit a bijection between them.

There is a bijection between V and the class of all ordinal numbers.

Bijection from to if is both an injection and a surjection.

The bijection is then called the isomorphism of the graphs.

For each pair pick a fixed bijection from onto .

If the function is a bijection any image can be inversely transformed.

The two coincide only if f is a bijection.

Now, another application of (A4) shows that there exists a bijection .

Accordingly, we can define two sets to "have the same number of elements" if there is a bijection between them.

A permutation is simply a bijection from the set of positive integers to itself.

Two sets have the same cardinality if and only if there is a bijection between them.

An isomorphism between linear orders is simply a strictly increasing bijection.

Two classes are equinumerous iff a bijection exists between them.

With that restriction there is an order-preserving bijection between the strings and the numbers.

With every bijection from the space to itself two coordinate transformations can be associated:

Where it is defined, the mapping is smooth function and bijection.

There is a bijection between distribution functions and characteristic functions.

However this reasoning is not constructive, as it still does not construct the required bijection.

This mapping gives a bijection between and the basis vectors .

One-to-one correspondence is an alternate name for a bijection.

A derangement of a set A is a bijection from A into itself that has no fixed points.

The bijection stated above between probability distributions and characteristic functions is continuous.

Then, the inner condition requires a bijection property from endomorphisms also to arrays.

By our initial assumption, , thus there exists a bijection .