What is the Proof of Equality for Sets of Random Variables?

[Kuma]

Hey there. I'm asked to prove:

If X1,...Xn are random variables defined on a set Ω and B1,...,Bn C R1 then prove that
(X1,...,Xn)^-1 (B1x...xBn) = X1^-1B1 n X2^-1B2 n ... n Xn^-1Bn

so I think I can explain the proof, but just not write it out. This is my attempt
if omega is on the left hand side:

the inverse image of X1^-1B1 is defined as all w in Ω such that X1(w) E B1 and that would apply to all Xi, Bi.
So the intersection of those events would be defined as all the w in Ω that
(X1,...,Xn)(w) E (B1x...xBn) which is the left hand side.

 

[Valkarie]

Now if omega is on the right hand side:the inverse image of X1^-1B1 is defined as all b in B1 such that X1^-1(b) E Ω and that would apply to all Xi, Bi. So the intersection of those events would be defined as all the b in (B1x...xBn) such that(X1^-1B1 n X2^-1B2 n ... n Xn^-1Bn)(b) E Ω which is the right hand side. Therefore, it can be concluded that (X1,...,Xn)^-1 (B1x...xBn) = X1^-1B1 n X2^-1B2 n ... n Xn^-1Bn