set h = f [#] c;
A1:
dom (f [#] c) = dom f
by Def39;
rng (f [#] c) c= N_PFuncs (DOMS Y)
proof
let y be
object ;
TARSKI:def 3 ( not y in rng (f [#] c) or y in N_PFuncs (DOMS Y) )
assume
y in rng (f [#] c)
;
y in N_PFuncs (DOMS Y)
then consider x being
object such that A2:
x in dom (f [#] c)
and A3:
(f [#] c) . x = y
by FUNCT_1:def 3;
reconsider y =
y as
Function by A3;
A4:
(f [#] c) . x = c (#) (f . x)
by A2, Def39;
A5:
rng y c= NAT
by A3, A4, ORDINAL1:def 12;
f . x in Y
by A1, A2, PARTFUN1:4;
then
dom (f . x) in { (dom m) where m is Element of Y : verum }
;
then A6:
dom (f . x) c= DOMS Y
by ZFMISC_1:74;
dom y = dom (f . x)
by A3, A4, VALUED_1:def 5;
then
y is
PartFunc of
(DOMS Y),
NAT
by A6, A5, RELSET_1:4;
hence
y in N_PFuncs (DOMS Y)
by Def18;
verum
end;
hence
f [#] c is PartFunc of X,(N_PFuncs (DOMS Y))
by A1, RELSET_1:4; verum