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 (([0] I) +* (i .--> x)) = I by A2, PZFMISC1:1;
then reconsider X = ([0] I) +* (i .--> x) as ManySortedSet of I by PARTFUN1:def 4, RELAT_1:def 18;
X is Element of bool M by Lm2, MSSUBFAM:11;
then X c= P .. X by A1, Def2;
then A4: X . i c= (P .. X) . i by A2, PBOOLE:def 5;
( dom (i .--> x) = {i} & i in {i} ) by FUNCOP_1:19, TARSKI:def 1;
then A5: X . i = (i .--> x) . i by FUNCT_4:14
.= x by FUNCOP_1:87 ;
i in dom P by A2, PARTFUN1:def 4;
hence x c= f . x by A3, A5, A4, PRALG_1:def 17; :: thesis: verum