Help with proving some facts

Hi,

I have to prove the following facts, but I am not sure if I figured them out correctly:

f : X --> Y and F : P(X) --> P(Y )

1) **If A1 is a subset of A2, then F(A1) is a subset of F(A2**)

For this one, I have the following:

For all x that are elements of A1 and A2

(there is a y that is an element of F(A1) s.t. F(x)=y) __and __(y is an element of F(A2) s.t. F(X)=y)

Therefore,

the fact that x is an element of A1 implies that there is a y that is an element of F(A2) s.t. F(x) = y

Hence, F(A1) is a subset of F(A2)

(Here I feel that, I missed a step constructing the proof with quantifiers.)

Also,

2) **For every A that is an element of P(X), A is a subset of the inverse function of F(A)**

For this one, I have:

(For all x that are elements of A, there is a y that is an element of F(A) s.t. F(x)=y s.t. the inverse of F(y)=x) __implies that__ (for all x belonging to A, x is an element of the inverse of F(F(A))) __implies that__ (A belongs to the inverse of F(F(A))).

I am also not sure about my logic here, do you guys agree with me?