let X be set ; :: thesis: ( dom (id X) = X & rng (id X) = X )
thus dom (id X) = X :: thesis: rng (id X) = X

proof

thus dom (id X) c= X :: according to XBOOLE_0:def 10 :: thesis: X c= dom (id X)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in dom (id X) or x in X )
assume x in dom (id X) ; :: thesis: x in X
then ex y being set st [x,y] in id X by Def4;
hence x in X by Def10; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in dom (id X) )
assume x in X ; :: thesis: x in dom (id X)
then [x,x] in id X by Def10;
hence x in dom (id X) by Def4; :: thesis: verum

end;

thus rng (id X) c= X :: according to XBOOLE_0:def 10 :: thesis: X c= rng (id X)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (id X) or x in X )
assume x in rng (id X) ; :: thesis: x in X
then consider y being set such that
A1: [y,x] in id X by Def5;
x = y by A1, Def10;
hence x in X by A1, Def10; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in rng (id X) )
( [x,x] in id X iff x in X ) by Def10;
hence ( not x in X or x in rng (id X) ) by Def5; :: thesis: verum