thus ( P = R implies for x being Element of X
for y being Element of Y holds
( [x,y] in P iff [x,y] in R ) ) ; :: thesis: ( ( for x being Element of X
for y being Element of Y holds
( [x,y] in P iff [x,y] in R ) ) implies P = R )
assume A1: for x being Element of X
for y being Element of Y holds
( [x,y] in P iff [x,y] in R ) ; :: thesis: P = R
let a, b be object ; :: according to RELAT_1:def 2 :: thesis: ( ( not [a,b] in P or [a,b] in R ) & ( not [a,b] in R or [a,b] in P ) )
hereby :: thesis: ( not [a,b] in R or [a,b] in P )
assume A2: [a,b] in P ; :: thesis: [a,b] in R
then reconsider a9 = a as Element of X by ZFMISC_1:87;
reconsider b9 = b as Element of Y by A2, ZFMISC_1:87;
[a9,b9] in R by A1, A2;
hence [a,b] in R ; :: thesis: verum
end;
assume A3: [a,b] in R ; :: thesis: [a,b] in P
then reconsider a9 = a as Element of X by ZFMISC_1:87;
reconsider b9 = b as Element of Y by A3, ZFMISC_1:87;
[a9,b9] in P by A1, A3;
hence [a,b] in P ; :: thesis: verum