let FT be non empty RelStr ; :: thesis: for x being Element of FT
for A being Subset of FT holds
( x in A ^f iff ex y being Element of FT st
( y in A & x in U_FT y ) )
let x be Element of FT; :: thesis: for A being Subset of FT holds
( x in A ^f iff ex y being Element of FT st
( y in A & x in U_FT y ) )
let A be Subset of FT; :: thesis: ( x in A ^f iff ex y being Element of FT st
( y in A & x in U_FT y ) )
thus ( x in A ^f implies ex y being Element of FT st
( y in A & x in U_FT y ) ) :: thesis: ( ex y being Element of FT st
( y in A & x in U_FT y ) implies x in A ^f )

proof

assume x in A ^f ; :: thesis: ex y being Element of FT st
( y in A & x in U_FT y )
then ex y being Element of FT st
( y = x & ex z being Element of FT st
( z in A & y in U_FT z ) ) ;
hence ex y being Element of FT st
( y in A & x in U_FT y ) ; :: thesis: verum

end;

assume ex z being Element of FT st
( z in A & x in U_FT z ) ; :: thesis: x in A ^f
hence x in A ^f ; :: thesis: verum