Definition of a filter
Let 
be a set.
Traditional definition of a filter on : A collection 
of subsets of is a filter on iff 
; and 
; and 
.
Alternative definition: A collection of subsets of is a filter on iff ; and 
.
The two definitions are equivalent.
Definition of the image of a filter
Let and 
be sets, a filter on and 
.
Traditional definition: The filter on image of by 
is 
.
Alternative definition: The filter on image of by is 
.
Again, the two definitions are equivalent.