let X1, X2 be set ; :: thesis: ( ( for x being set holds
( x in X1 iff ex y being set st [x,y] in R ) ) & ( for x being set holds
( x in X2 iff ex y being set st [x,y] in R ) ) implies X1 = X2 )
assume that
A2: for x being set holds
( x in X1 iff ex y being set st [x,y] in R ) and
A3: for x being set holds
( x in X2 iff ex y being set st [x,y] in R ) ; :: thesis: X1 = X2

now

let x be set ; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff ex y being set st [x,y] in R ) by A2;
hence ( x in X1 iff x in X2 ) by A3; :: thesis: verum

end;

hence X1 = X2 by TARSKI:2; :: thesis: verum