There is something that I'm not getting about functions and cardinality of sets.

I've read the following:

If $F$ is a one to one function, then $F^{-1}$ (the inverse) is also a one-to-one function.

Because when I read about cardinality of sets, I encounter with the definition:

$|A|\leq |\mathcal P(A)|$ if there is a one to one function from $A$ to $\mathcal P(A)$.

This function is trivial $F(a) = \{a\}$ and it's one to one and not onto.

But if I follow the statement from above of $F^{-1}$ (the inverse of $F$) being also one to one, this leads me to the following :

$|\mathcal P(A)|\leq |A|$

Which I know is false. So there is obviously something wrong with my understanding, what concept am I missing, what I'm getting wrong?