let A, B, C, D, E, F, J be set ; :: thesis: for h being Function
for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds
rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)}
let h be Function; :: thesis: for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds
rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)}
let A9, B9, C9, D9, E9, F9, J9 be set ; :: thesis: ( h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} )
assume h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) ; :: thesis: rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)}
then A1: dom h = {A,B,C,D,E,F,J} by Th50;
then B in dom h by ENUMSET1:def 5;
then A2: h . B in rng h by FUNCT_1:def 3;
F in dom h by A1, ENUMSET1:def 5;
then A3: h . F in rng h by FUNCT_1:def 3;
E in dom h by A1, ENUMSET1:def 5;
then A4: h . E in rng h by FUNCT_1:def 3;
D in dom h by A1, ENUMSET1:def 5;
then A5: h . D in rng h by FUNCT_1:def 3;
C in dom h by A1, ENUMSET1:def 5;
then A6: h . C in rng h by FUNCT_1:def 3;
J in dom h by A1, ENUMSET1:def 5;
then A7: h . J in rng h by FUNCT_1:def 3;
A8: rng h c= {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)}

proof

let t be object ; :: according to TARSKI:def 3 :: thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} )
assume t in rng h ; :: thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)}
then consider x1 being object such that
A9: x1 in dom h and
A10: t = h . x1 by FUNCT_1:def 3;
( x1 = A or x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J ) by A1, A9, ENUMSET1:def 5;
hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} by A10, ENUMSET1:def 5; :: thesis: verum

end;

A in dom h by A1, ENUMSET1:def 5;
then A11: h . A in rng h by FUNCT_1:def 3;
{(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} c= rng h by A11, A2, A6, A5, A4, A3, A7, ENUMSET1:def 5;
hence rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} by A8, XBOOLE_0:def 10; :: thesis: verum