let Y1, Y2 be set ; :: thesis: ( ( for y being set holds
( y in Y1 iff ex x being set st [x,y] in R ) ) & ( for y being set holds
( y in Y2 iff ex x being set st [x,y] in R ) ) implies Y1 = Y2 )
assume A2: ( ( for y being set holds
( y in Y1 iff ex x being set st [x,y] in R ) ) & ( for y being set holds
( y in Y2 iff ex x being set st [x,y] in R ) ) ) ; :: thesis: Y1 = Y2

now

let y be set ; :: thesis: ( y in Y1 iff y in Y2 )
( ( y in Y1 implies ex x being set st [x,y] in R ) & ( ex x being set st [x,y] in R implies y in Y1 ) & ( y in Y2 implies ex x being set st [x,y] in R ) & ( ex x being set st [x,y] in R implies y in Y2 ) ) by A2;
hence ( y in Y1 iff y in Y2 ) ; :: thesis: verum

end;

hence Y1 = Y2 by TARSKI:2; :: thesis: verum