let C, D, E be non empty set ; for f being Function of [:C,D:],E holds curry f is Function of C,(Funcs (D,E))
let f be Function of [:C,D:],E; curry f is Function of C,(Funcs (D,E))
A1:
dom f = [:C,D:]
by FUNCT_2:def 1;
A2:
rng (curry f) c= Funcs (D,E)
proof
A3:
rng (curry f) c= Funcs (
D,
(rng f))
by A1, Th28;
let x be
object ;
TARSKI:def 3 ( not x in rng (curry f) or x in Funcs (D,E) )
assume
x in rng (curry f)
;
x in Funcs (D,E)
then consider g being
Function such that A4:
x = g
and A5:
dom g = D
and A6:
rng g c= rng f
by A3, FUNCT_2:def 2;
rng g c= E
by A6, XBOOLE_1:1;
then
g is
Function of
D,
E
by A5, FUNCT_2:def 1, RELSET_1:4;
hence
x in Funcs (
D,
E)
by A4, FUNCT_2:8;
verum
end;
dom (curry f) = C
by A1, Th17;
hence
curry f is Function of C,(Funcs (D,E))
by A2, FUNCT_2:def 1, RELSET_1:4; verum