let T be non empty TopSpace; :: thesis: for s being Function of , the carrier of T holds filter_image (s,) = { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A }
let s be Function of , the carrier of T; :: thesis: filter_image (s,) = { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A }
set X = { M where M is Subset of the carrier of T : s " M in Frechet_Filter } ;
set Y = { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } ;
{ M where M is Subset of the carrier of T : s " M in Frechet_Filter } = { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A }
proof
now :: thesis: for x being object st x in { M where M is Subset of the carrier of T : s " M in Frechet_Filter } holds
x in { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A }
let x be object ; :: thesis: ( x in { M where M is Subset of the carrier of T : s " M in Frechet_Filter } implies x in { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } )
assume x in { M where M is Subset of the carrier of T : s " M in Frechet_Filter } ; :: thesis: x in { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A }
then consider M being Subset of the carrier of T such that
A1: x = M and
A2: s " M in Frechet_Filter ;
ex A being finite Subset of st s " M = \ A by ;
hence x in { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } by A1; :: thesis: verum
end;
then A3: { M where M is Subset of the carrier of T : s " M in Frechet_Filter } c= { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } ;
now :: thesis: for x being object st x in { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } holds
x in { M where M is Subset of the carrier of T : s " M in Frechet_Filter }
let x be object ; :: thesis: ( x in { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } implies x in { M where M is Subset of the carrier of T : s " M in Frechet_Filter } )
assume x in { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } ; :: thesis: x in { M where M is Subset of the carrier of T : s " M in Frechet_Filter }
then consider M being Subset of the carrier of T such that
A4: x = M and
A5: ex A being finite Subset of st s " M = \ A ;
s " M in Frechet_Filter by ;
hence x in { M where M is Subset of the carrier of T : s " M in Frechet_Filter } by A4; :: thesis: verum
end;
then { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } c= { M where M is Subset of the carrier of T : s " M in Frechet_Filter } ;
hence { M where M is Subset of the carrier of T : s " M in Frechet_Filter } = { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } by A3; :: thesis: verum
end;
hence filter_image (s,) = { M where M is Subset of the carrier of T : ex A being finite Subset of st s " M = \ A } ; :: thesis: verum