let A, B, C, D, E, F be set ; :: thesis: for h being Function
for A', B', C', D', E', F' being set st h = (((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (A .--> A') holds
dom h = {A,B,C,D,E,F}
let h be Function; :: thesis: for A', B', C', D', E', F' being set st h = (((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (A .--> A') holds
dom h = {A,B,C,D,E,F}
let A', B', C', D', E', F' be set ; :: thesis: ( h = (((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (A .--> A') implies dom h = {A,B,C,D,E,F} )
assume A1: h = (((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (A .--> A') ; :: thesis: dom h = {A,B,C,D,E,F}
A2: dom (((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) = {F,B,C,D,E} by Th30
.= {F} \/ {B,C,D,E} by ENUMSET1:47
.= {B,C,D,E,F} by ENUMSET1:50 ;
dom (A .--> A') = {A} by FUNCOP_1:19;
then dom ((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (A .--> A')) = {B,C,D,E,F} \/ {A} by A2, FUNCT_4:def 1
.= {A,B,C,D,E,F} by ENUMSET1:51 ;
hence dom h = {A,B,C,D,E,F} by A1; :: thesis: verum