let X, Y be set ; :: thesis: for f being Function st dom f c= X & rng f c= Y holds
f = <:f,X,Y:>
let f be Function; :: thesis: ( dom f c= X & rng f c= Y implies f = <:f,X,Y:> )
assume A1: ( dom f c= X & rng f c= Y ) ; :: thesis: f = <:f,X,Y:>
A2: dom f c= dom <:f,X,Y:>

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in dom f or x in dom <:f,X,Y:> )
assume A3: x in dom f ; :: thesis: x in dom <:f,X,Y:>
then f . x in rng f by FUNCT_1:def 3;
hence x in dom <:f,X,Y:> by A1, A3, Th24; :: thesis: verum

end;

dom <:f,X,Y:> c= dom f by Th23;
then A4: dom f = dom <:f,X,Y:> by A2;
for x being object st x in dom f holds
f . x = <:f,X,Y:> . x by A2, Th26;
hence f = <:f,X,Y:> by A4; :: thesis: verum