I had recently gotten into an argument with a fellow player on the subject of how spellbooks work. Specifically, does a Wizard require access to his book in order to cast spells, or just to prepare them? In context we were facing against an enemy party that included a wizard, and the player mentioned suggested trying to shoot the Wizard's spellbook using his bow in order to sunder it and, he assumed, disable all her prepared spells. I and some other players contested that and in the end he decided to just shoot her in the head instead, but just to double check: would that have worked or not?
Just For Preparation
The short version is: you're right. Wizards need a spellbook to prepare spells (from the book), but not to cast. The book is not a component/material/focus/requirement of casting the spell.
The preparing spells rules are here, and to prepare from a spellbook you need the book. That part is pretty straightforward. Here's an important part:
Until she prepares spells from her spellbook, the only spells a wizard has available to cast are the ones that she already had prepared from the previous day and has not yet used.
The casting rules are here, including what you need. Here's one useful note:
First you must choose which spell to cast. If you’re a cleric, druid, experienced paladin, experienced ranger, or wizard, you select from among spells prepared earlier in the day and not yet cast
You'll note that in all the rules for casting a spell, the word "book" never appears. Wizards can cast anything they have prepared, provided they have the requirements listed in the spell. Those include things like material components, focus, the ability to speak or do somatic gestures, and anything else that a spell might list. Spellbooks are never listed for the Wizard class.
Spellbook to prepare is correct. Imagine the spellbook like a cookbook. You know what the recipe is, but you can't really remember every single detail, so before cooking, you look over the recipe book, and refresh your memory, then you cook your dish.
The Spell Mastery feat will allow a certain number of spells to no longer need their spellbook to prepare that particular spell. Imagine that same chef, who doesn't need to look at his cookbook anymore for a particular dish. He has cooked it so many times he can recall it from memory. Now he will stop to think every now and then, but it is so deeply engrained in his memory, he doesn't need to consult his cookbook anymore.