Here's why:
* The "i" is followed by a consonant ("br"): In general, when a vowel is followed by a consonant, it tends to have a short sound.
* The "y" acts as a consonant in this case: The "y" at the end of the word "library" functions as a consonant, not a vowel.
So, the word "library" has a short "i" sound because of the vowel-consonant pattern and the consonant function of the "y."