let X0, Y0 be finite natural-membered set ; :: thesis: for i being Nat st X0 <N< Y0 & i < card X0 holds
( rng ((Sgm0 (X0 \/ Y0)) | (card X0)) = X0 & ((Sgm0 (X0 \/ Y0)) | (card X0)) . i = (Sgm0 (X0 \/ Y0)) . i )
let i be Nat; :: thesis: ( X0 <N< Y0 & i < card X0 implies ( rng ((Sgm0 (X0 \/ Y0)) | (card X0)) = X0 & ((Sgm0 (X0 \/ Y0)) | (card X0)) . i = (Sgm0 (X0 \/ Y0)) . i ) )
assume A1: ( X0 <N< Y0 & i < card X0 ) ; :: thesis: ( rng ((Sgm0 (X0 \/ Y0)) | (card X0)) = X0 & ((Sgm0 (X0 \/ Y0)) | (card X0)) . i = (Sgm0 (X0 \/ Y0)) . i )
A20: X0 \/ Y0 <> {} by A1, CARD_1:47, XBOOLE_1:15;
set f0 = Sgm0 (X0 \/ Y0);
A30: rng (Sgm0 (X0 \/ Y0)) = X0 \/ Y0 by AC113;
set f = (Sgm0 (X0 \/ Y0)) | (card X0);
set Z = { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } ;
A2: dom ((Sgm0 (X0 \/ Y0)) | (card X0)) = len ((Sgm0 (X0 \/ Y0)) | (card X0)) ;
A33: len (Sgm0 (X0 \/ Y0)) = card (X0 \/ Y0) by Th44;
A22n: X0 c= X0 \/ Y0 by XBOOLE_1:7;
A22: card X0 <= len (Sgm0 (X0 \/ Y0)) by A33, NAT_1:44, XBOOLE_1:7;
A4: len ((Sgm0 (X0 \/ Y0)) | (card X0)) = card X0 by TH80, A33, A22n, NAT_1:44;
A4d: len (Sgm0 (X0 \/ Y0)) = card (X0 \/ Y0) by Th44;
B4: rng ((Sgm0 (X0 \/ Y0)) | (card X0)) c= rng (Sgm0 (X0 \/ Y0)) by RELAT_1:99;
A3: rng ((Sgm0 (X0 \/ Y0)) | (card X0)) c= { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) }
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((Sgm0 (X0 \/ Y0)) | (card X0)) or y in { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } )
assume B1: y in rng ((Sgm0 (X0 \/ Y0)) | (card X0)) ; :: thesis: y in { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) }
then consider x being set such that
B2: ( x in dom ((Sgm0 (X0 \/ Y0)) | (card X0)) & y = ((Sgm0 (X0 \/ Y0)) | (card X0)) . x ) by FUNCT_1:def 5;
reconsider nx = x as Nat by B2;
B3: nx in card X0 by B2, TH80, A33, A22n, A2, NAT_1:44;
reconsider y0 = y as Element of X0 \/ Y0 by B1, B4, AC113;
ex k2 being Nat st
( y0 = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) by B3, B2;
hence y in { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } ; :: thesis: verum
end;
A40n: { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } c= rng ((Sgm0 (X0 \/ Y0)) | (card X0))
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } or y in rng ((Sgm0 (X0 \/ Y0)) | (card X0)) )
assume y in { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } ; :: thesis: y in rng ((Sgm0 (X0 \/ Y0)) | (card X0))
then consider v0 being Element of X0 \/ Y0 such that
B2: ( y = v0 & ex k2 being Nat st
( v0 = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) ) ;
thus y in rng ((Sgm0 (X0 \/ Y0)) | (card X0)) by B2, A4, FUNCT_1:def 5; :: thesis: verum
end;
then A40: rng ((Sgm0 (X0 \/ Y0)) | (card X0)) = { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } by A3, XBOOLE_0:def 10;
A50n: now
assume B1: ( not { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } c= X0 & not X0 c= { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } ) ; :: thesis: contradiction
then consider v1 being set such that
B2: ( v1 in { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } & not v1 in X0 ) by TARSKI:def 3;
consider v2 being set such that
B3: ( v2 in X0 & not v2 in { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } ) by B1, TARSKI:def 3;
consider v10 being Element of X0 \/ Y0 such that
B4: ( v1 = v10 & ex k2 being Nat st
( v10 = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) ) by B2;
B98: v10 in Y0 by A20, B2, B4, XBOOLE_0:def 3;
consider k20 being Nat such that
B4b: ( v10 = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k20 & k20 in card X0 ) by B4;
reconsider nv10 = v10 as Nat ;
X0 c= X0 \/ Y0 by XBOOLE_1:7;
then consider x2 being set such that
B5: ( x2 in dom (Sgm0 (X0 \/ Y0)) & v2 = (Sgm0 (X0 \/ Y0)) . x2 ) by B3, A30, FUNCT_1:def 5;
reconsider x20 = x2 as Nat by B5;
reconsider nv2 = v2 as Nat by B5;
B61n: now
assume C0: x20 < card X0 ; :: thesis: contradiction
card X0 <= card (X0 \/ Y0) by NAT_1:44, XBOOLE_1:7;
then C4: ( card X0 <= len (Sgm0 (X0 \/ Y0)) & x20 in card X0 ) by Th44, C0, NAT_1:45;
then ((Sgm0 (X0 \/ Y0)) | (card X0)) . x20 = (Sgm0 (X0 \/ Y0)) . x20 by Th19;
hence contradiction by B3, B5, C4, A22n; :: thesis: verum
end;
k20 < len ((Sgm0 (X0 \/ Y0)) | (card X0)) by B4b, A4, NAT_1:45;
then B62: k20 < x20 by B61n, A4, XXREAL_0:2;
B66: card X0 <= len (Sgm0 (X0 \/ Y0)) by A4d, NAT_1:44, XBOOLE_1:7;
B44: ((Sgm0 (X0 \/ Y0)) | (card X0)) . k20 = (Sgm0 (X0 \/ Y0)) . k20 by Th19, B66, B4b;
x20 < len (Sgm0 (X0 \/ Y0)) by B5, NAT_1:45;
then nv10 < nv2 by B4b, B5, AC113, B62, B44;
hence contradiction by B3, AE100, A1, B98; :: thesis: verum
end;
(Sgm0 (X0 \/ Y0)) | (card X0) is one-to-one by FUNCT_1:84;
then A51: dom ((Sgm0 (X0 \/ Y0)) | (card X0)),((Sgm0 (X0 \/ Y0)) | (card X0)) .: (dom ((Sgm0 (X0 \/ Y0)) | (card X0))) are_equipotent by CARD_1:60;
((Sgm0 (X0 \/ Y0)) | (card X0)) .: (dom ((Sgm0 (X0 \/ Y0)) | (card X0))) = rng ((Sgm0 (X0 \/ Y0)) | (card X0)) by RELAT_1:146;
then card { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } = card (len ((Sgm0 (X0 \/ Y0)) | (card X0))) by A40, A51, CARD_1:21;
then A53: card { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } = card X0 by TH80, A33, A22n, NAT_1:44;
then X0,{ v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } are_equipotent by CARD_1:21;
then reconsider Z0 = { v where v is Element of X0 \/ Y0 : ex k2 being Nat st
( v = ((Sgm0 (X0 \/ Y0)) | (card X0)) . k2 & k2 in card X0 ) } as finite set by CARD_1:68;
A77n: now
per cases ( Z0 c= X0 or X0 c= Z0 ) by A50n;
case Z0 c= X0 ; :: thesis: Z0 = X0
hence Z0 = X0 by A53, CARD_FIN:1; :: thesis: verum
end;
case X0 c= Z0 ; :: thesis: Z0 = X0
hence Z0 = X0 by A53, CARD_FIN:1; :: thesis: verum
end;
end;
end;
( card X0 <= len (Sgm0 (X0 \/ Y0)) & i in card X0 ) by A22, A1, NAT_1:45;
hence ( rng ((Sgm0 (X0 \/ Y0)) | (card X0)) = X0 & ((Sgm0 (X0 \/ Y0)) | (card X0)) . i = (Sgm0 (X0 \/ Y0)) . i ) by A3, A40n, A77n, Th19, XBOOLE_0:def 10; :: thesis: verum