The resolution of this apparent paradox is that, while this conclusion is true internally, it is not true externally: namely any two infinite sets are countable externally in that model, hence there is a $1$$1$ function between any two of them including for a model of some uncountable set $X$ and of its power set $P(X)$. However, the that function (or its graph) is not in the model! One can enlarge the model by adding that function (and more). But this extended model will necessary have $P(X)$ uncountable externally and there is no $1$$1$ function from $X$ to $P(X)$ externally any more.