Anesthesia Consents- question

Hi...I have a question. Our anesthesiologists at our hospital always talk to the patients pre op to get a thorough H&P. However, when they get the pt to sign the anesthesia consent they always tell them what they will be doing for them intra op/ and post op. But...when the procedure starts they often times get another anesthesiologist to do the case or get a SRNA to do the case and they do paperwork in the office and only wander in for the intubation and induction and extubation. The patients belive the anesthesiologist that got them to sign the consent will be present and managing their entire case. My question is this: When a anesthesiologist gets a patient to sign a consent for the surgery do they have the obligation to physcially do their case or can they substitute another anesthesia MD ....or even get a student SRNA do the case without the patients knowledge?

I cant help but think of it like this- If I get a cardiothoracic sugeon to do my open heart surgery, and that CT MD gets me to sign a consent for the surgery.....then to me ....that MD better be doing my surgery.....or else my consent is invalid . But with anesthesia is it the same ....what is ok ?