The purchase and your mortgage are essentially two separate matters.They are related insofar as you should not sign contracts for the purchase until you know for definite that you can proceed with the purchase of the property from the vendor because you definitely have the funds to complete. But if you have an offer of a mortgage from one lender then you can sign the contracts. You could still take the offer of a mortgage from another lender after signing the contracts.
Increasingly, I am advising clients to make a purchase contract conditional on the funds being available at closing. This is great from a purchasers point of view but useless for a vendor as the contracts are always conditional until closing. The whole thing can collapse right up until the day of closing.
mf