let Y1, Y2 be set ; :: thesis: ( ( for y being set holds
( y in Y1 iff ex x being set st
( [x,y] in R & x in X ) ) ) & ( for y being set holds
( y in Y2 iff ex x being set st
( [x,y] in R & x in X ) ) ) implies Y1 = Y2 )
assume that
A4: for y being set holds
( y in Y1 iff ex x being set st
( [x,y] in R & x in X ) ) and
A5: for y being set holds
( y in Y2 iff ex x being set st
( [x,y] in R & x in X ) ) ; :: thesis: Y1 = Y2

now

let y be set ; :: thesis: ( y in Y1 iff y in Y2 )
( y in Y1 iff ex x being set st
( [x,y] in R & x in X ) ) by A4;
hence ( y in Y1 iff y in Y2 ) by A5; :: thesis: verum

end;

hence Y1 = Y2 by TARSKI:1; :: thesis: verum