SQL UPDATE and DELETE with aliases not working

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SQL

Hello folks.

Can anyone tell me what is wrong with the following SQL statements?

The purpose purpose of the UPDATE is to promote one of the prod_cat_code values for a product that are stored in the product_category_xref table up to the product table (any value will be good - even NULL). So I have:

UPDATE p 
SET    p.prod_cat_code = (SELECT TOP(1) pcx.cat_code 
                          FROM          prod_category_xref pcx 
                          WHERE         pcx.prod_num = p.prod_num) 
FROM   product p; 

After that, I want to get rid any records in product_category_xref which have the same prod_cat_code as the one now stored on the product record.

DELETE pcx 
FROM   [dbo].[product_category_xref] pcx 
WHERE  EXISTS (SELECT [dbo].[product] AS p 
               WHERE  p.prod_num = pcx.prod_num 
                      AND p.prod_cat_code = pcx.prod_cat_code);

However, when I execute these two statements I get errors for the DELETE:

Msg 4104, Level 16, State 1, Line 1
The multi-part identifier "p.prod_num" could not be bound.
Msg 4104, Level 16, State 1, Line 1
The multi-part identifier "p.prod_cat_code" could not be bound.
Msg 4104, Level 16, State 1, Line 1
The multi-part identifier "dbo.product" could not be bound.

I don't understand why. I eventually changed the DELETE to

DELETE pcx 
FROM   product_category_xref pcx 
       JOIN product AS p 
            ON p.prod_num = pcx.prod_Num 
               AND p.prod_cat_code = pcx.prod_cat_code;

which accomplishes the same thing. But now I get an error on the UPDATE (which didn't throw an error before):

Msg 208, Level 16, State 1, Line 9
Invalid object name 'prod_category_xref'.

All I can say is 'huh?' I know that I can change the UPDATE to work too, but I don't see why the original statements do not.
Any ideas?