let f1, f2 be Function; :: thesis: ( dom f1 = X & ( for x being set st x in X holds
f1 . x = [x,x] ) & dom f2 = X & ( for x being set st x in X holds
f2 . x = [x,x] ) implies f1 = f2 )
assume that
A1: dom f1 = X and
A2: for x being set st x in X holds
f1 . x = [x,x] and
A3: dom f2 = X and
A4: for x being set st x in X holds
f2 . x = [x,x] ; :: thesis: f1 = f2
for x being set st x in X holds
f1 . x = f2 . x

proof

let x be set ; :: thesis: ( x in X implies f1 . x = f2 . x )
assume A5: x in X ; :: thesis: f1 . x = f2 . x
then f1 . x = [x,x] by A2;
hence f1 . x = f2 . x by A4, A5; :: thesis: verum

end;

hence f1 = f2 by A1, A3, FUNCT_1:2; :: thesis: verum