I need to proof knowing that some two sets

and

have the same cardinality that sets

and

, which are sets of maps from some set

to sets

and

respectively, also have the same cardinality.

So here's my attempt:

So we know that those two sets are equal, so that means that there's a bijection between them, like this:

I have to create a bijection between

and

to prove that they have the same cardinality. So let

be this bijection.

is supposed to take a function that goes from

to

and return a function that goes from

to

.

for some

equals

meaning

.

Let us construct a function

that does this

. This function takes

from

and returns an element in

by assigning one to the result of

by the bijection between sets

and

.

So in the end, a bijection between sets

and

is a function

.