Factorization Theorem for Sufficient Statistics

Problem:

Let Y1,Y2,...,Yn denote a random sample from the uniform distribution over the interval (0,theta). Show that Y(n)=max(Y1,Y2,...,Yn) is a sufficient statistic for theta by the factorization theorem.

Solution:

http://www.geocities.com/asdfasdf23135/stat10.JPG

1) While I understand that I_A (x)I_B (x)=I_A intersect B (x), I don't understand the equality circled in red above.

In the solutions, they say that I_0,theta (y1)...I_ 0,theta(yn)=I_0,theta (y(n)). Is this really correct?

Shouldn't the right hand side be I_0,theta (y(n))I_0,infinity (y(1)) ? I believe that the second factor is necessary because the largest observation is greater than zero does not guarantee that the smallest observation is greater than zero.

Which one is correct?

2) Also, is I_0,theta (y(n)) a function of y(n), a function of theta, or a function of both y(n) and theta?

If it is a function of both y(n) and theta, then there is something that I don't understand. Following the definition of indicator function that I_A (x) is a function of x alone (it is a function of only the stuff in the parenthesis), shouldn't I_0,theta (y(n)) be a function of only y(n) alone?

Thank you for explaining! I've been confused with these ideas for at least a week.