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 set ; :: 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, Th78; :: thesis: verum

end;

dom <:f,X,Y:> c= dom f by Th77;
then A4: dom f = dom <:f,X,Y:> by A2, XBOOLE_0:def 10;
for x being set st x in dom f holds
f . x = <:f,X,Y:> . x by A2, Th80;
hence f = <:f,X,Y:> by A4, FUNCT_1:2; :: thesis: verum