let X be set ; :: thesis: for R being Relation of X st R is_reflexive_in X holds
dom R = X
let R be Relation of X; :: thesis: ( R is_reflexive_in X implies dom R = X )
assume A1: R is_reflexive_in X ; :: thesis: dom R = X
for x being object st x in X holds
ex y being object st [x,y] in R

proof

let x be object ; :: thesis: ( x in X implies ex y being object st [x,y] in R )
assume A2: x in X ; :: thesis: ex y being object st [x,y] in R
take x ; :: thesis: [x,x] in R
thus [x,x] in R by A1, A2, RELAT_2:def 1; :: thesis: verum

end;

hence dom R = X by RELSET_1:9; :: thesis: verum