let i, I be set ; :: thesis: for M being ManySortedSet of I
for f being Function
for P being MSSetOp of M st P is reflexive & i in I & f = P . i holds
for x being Element of bool (M . i) holds x c= f . x
let M be ManySortedSet of I; :: thesis: for f being Function
for P being MSSetOp of M st P is reflexive & i in I & f = P . i holds
for x being Element of bool (M . i) holds x c= f . x
let f be Function; :: thesis: for P being MSSetOp of M st P is reflexive & i in I & f = P . i holds
for x being Element of bool (M . i) holds x c= f . x
let P be MSSetOp of M; :: thesis: ( P is reflexive & i in I & f = P . i implies for x being Element of bool (M . i) holds x c= f . x )
assume that
A1: P is reflexive and
A2: i in I and
A3: f = P . i ; :: thesis: for x being Element of bool (M . i) holds x c= f . x
let x be Element of bool (M . i); :: thesis: x c= f . x
dom ((EmptyMS I) +* (i .--> x)) = I by A2, PZFMISC1:1;
then reconsider X = (EmptyMS I) +* (i .--> x) as ManySortedSet of I by PARTFUN1:def 2, RELAT_1:def 18;
X is Element of bool M by Lm2, MSSUBFAM:11;
then X c= P .. X by A1;
then A4: X . i c= (P .. X) . i by A2;
( dom (i .--> x) = {i} & i in {i} ) by TARSKI:def 1;
then A5: X . i = (i .--> x) . i by FUNCT_4:13
.= x by FUNCOP_1:72 ;
( i in dom X & i in dom P ) by A2, PARTFUN1:def 2;
then i in (dom P) /\ (dom X) by XBOOLE_0:def 4;
then i in dom (P .. X) by PRALG_1:def 19;
hence x c= f . x by A3, A5, A4, PRALG_1:def 19; :: thesis: verum