The text I'm reading proves uncountability of the Cantor's set by showing that the Cantor's function is a surjective map from the Cantor's set onto [0, 1]. 

I think that it can be shown by direct use of the Cantor's diagonal argument for the Cantor's set, i.e., without use of the Cantor's function.

Am I right or am I missing something?

Just now, Genady said:

The text I'm reading proves uncountability of the Cantor's set by showing that the Cantor's function is a surjective map from the Cantor's set onto [0, 1]. 

I think that it can be shown by direct use of the Cantor's diagonal argument for the Cantor's set, i.e., without use of the Cantor's function.

Am I right or am I missing something?

The [0,1] map is probably the standard way of doing this but there are others eg in ternary.