reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;
reserve n for Element of omega;
reserve e,u for set;

theorem Th78:
  for F,G be Function holds
  F,G are_fiberwise_equipotent iff for X be set holds card (F"X) = card (G"X)
proof
  let F,G be Function;
  thus F,G are_fiberwise_equipotent implies
  for X be set holds card(F"X) = card(G"X)
  proof
    assume F,G are_fiberwise_equipotent;
    then consider H be Function such that
A1: dom H = dom F and
A2: rng H = dom G and
A3: H is one-to-one and
A4: F = G*H by Th77;
    let X be set;
    set t = H|(F"X);
A5: t is one-to-one by A3,FUNCT_1:52;
A6: dom t = F"X by A1,RELAT_1:62,132;
 rng t = G"X
    proof
      thus rng t c= G"X
      proof
        let z be object;
        assume z in rng t;
        then consider y being object such that
A7:    y in dom t and
A8:    t.y = z by FUNCT_1:def 3;
A9:    F.y in X by A6,A7,FUNCT_1:def 7;
A10:    z = H.y by A7,A8,FUNCT_1:47;
    dom t = dom H /\ F"X by RELAT_1:61;
then A11:    y in dom H by A7,XBOOLE_0:def 4;
then A12:    z in dom G by A2,A10,FUNCT_1:def 3;
    G.z in X by A4,A9,A10,A11,FUNCT_1:13;
        hence thesis by A12,FUNCT_1:def 7;
      end;
      let z be object;
      assume
A13:  z in G"X;
then A14:  z in dom G by FUNCT_1:def 7;
A15:  G.z in X by A13,FUNCT_1:def 7;
      consider y being object such that
A16:  y in dom H and
A17:  H.y = z by A2,A14,FUNCT_1:def 3;
  F.y in X by A4,A15,A16,A17,FUNCT_1:13;
then A18:  y in dom t by A1,A6,A16,FUNCT_1:def 7;
then   t.y in rng t by FUNCT_1:def 3;
      hence thesis by A17,A18,FUNCT_1:47;
    end;
    hence thesis by CARD_1:5,A5,A6,WELLORD2:def 4;
  end;
  assume for X be set holds card(F"X) = card(G"X);
  hence for x being object holds card Coim(F,x) = card Coim(G,x);
end;
