
theorem Th19:
for X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
 B be Subset of Y st T is bijective holds T"B in S iff B in CopyField(T,S)
proof
    let X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
    B be Subset of Y;
    assume T is bijective; then
    consider H be Function of Y,X such that
A1: H is bijective & H = T" & H" = T & .:H = (.:T)"
  & (.:H).:CopyField(T,S) = S & CopyField(H,CopyField(T,S)) = S by Th17;

    B in CopyField(T,S) iff H.:B in CopyField(H,CopyField(T,S)) by Th18,A1;
    hence T"B in S iff B in CopyField(T,S) by A1,FUNCT_1:85;
end;
