let A, B, C be set ; :: thesis: ( ( not C = {} or B = {} or A = {} ) implies for f being Function of A,(Funcs (B,C)) holds rng (Frege f) c= Funcs (A,C) )
assume ( not C = {} or B = {} or A = {} ) ; :: thesis: for f being Function of A,(Funcs (B,C)) holds rng (Frege f) c= Funcs (A,C)
then A1: ( Funcs (B,C) = {} implies A = {} ) by FUNCT_2:11;
let f be Function of A,(Funcs (B,C)); :: thesis: rng (Frege f) c= Funcs (A,C)
A2: SubFuncs (rng f) = rng f by Th4;
then A3: dom (rngs f) = f " (rng f) by FUNCT_6:def 3;
then A4: dom (rngs f) = dom f by RELAT_1:169
.= A by A1, FUNCT_2:def 1 ;
A5: for x being set st x in dom (rngs f) holds
(rngs f) . x c= (A --> C) . x

proof

let x be set ; :: thesis: ( x in dom (rngs f) implies (rngs f) . x c= (A --> C) . x )
assume A6: x in dom (rngs f) ; :: thesis: (rngs f) . x c= (A --> C) . x
A7: (rngs f) . x = proj2 (f . x) by A2, A3, A6, FUNCT_6:def 3
.= rng (f . x) ;
f . x in Funcs (B,C) by A1, A4, A6, FUNCT_2:7;
then (rngs f) . x c= C by A7, FUNCT_2:169;
hence (rngs f) . x c= (A --> C) . x by A4, A6, FUNCOP_1:13; :: thesis: verum

end;

dom (rngs f) = dom (A --> C) by A4, FUNCOP_1:19;
then product (rngs f) c= product (A --> C) by A5, CARD_3:38;
then product (rngs f) c= Funcs (A,C) by CARD_3:20;
hence rng (Frege f) c= Funcs (A,C) by FUNCT_6:58; :: thesis: verum