Expectation conditional on indicator function

Let T and K be dependent continuous random variables, and note the Indicator function as I{.}:

Is it correct to say that $E[T|I\{T>t,K<k\}]=E[T|T>t,K<k]$? Is that a property of the Indicator function?

via Recent Questions - Mathematics - Stack Exchange http://math.stackexchange.com/questions/289664/expectation-conditional-on-indicator-function