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 set st x in X holds
ex y being set st [x,y] in R

proof

let x be set ; :: thesis: ( x in X implies ex y being set st [x,y] in R )
assume A2: x in X ; :: thesis: ex y being set st [x,y] in R
take x1 = x; :: thesis: [x,x1] in R
thus [x,x1] in R by A1, A2, RELAT_2:def 1; :: thesis: verum

end;

hence dom R = X by RELSET_1:22; :: thesis: verum