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Theorem isores1 5593
Description: An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
isores1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
) )

Proof of Theorem isores1
StepHypRef Expression
1 isocnv 5590 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 isores2 5592 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  <->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A ) ) ( B ,  A ) )
31, 2sylib 120 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A
) ) ( B ,  A ) )
4 isocnv 5590 . . . 4  |-  ( `' H  Isom  S , 
( R  i^i  ( A  X.  A ) ) ( B ,  A
)  ->  `' `' H  Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
) )
53, 4syl 14 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' `' H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
6 isof1o 5586 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
7 f1orel 5256 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Rel  H )
8 dfrel2 4881 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
9 isoeq1 5580 . . . . 5  |-  ( `' `' H  =  H  ->  ( `' `' H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B )  <->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) ) )
108, 9sylbi 119 . . . 4  |-  ( Rel 
H  ->  ( `' `' H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B )  <->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) ) )
116, 7, 103syl 17 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( `' `' H  Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  <->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) ) )
125, 11mpbid 145 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
13 isocnv 5590 . . . . 5  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A ) ) ( B ,  A ) )
1413, 2sylibr 132 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' H  Isom  S ,  R  ( B ,  A ) )
15 isocnv 5590 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' `' H  Isom  R ,  S  ( A ,  B ) )
1614, 15syl 14 . . 3  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' `' H  Isom  R ,  S  ( A ,  B ) )
17 isof1o 5586 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H : A
-1-1-onto-> B )
18 isoeq1 5580 . . . . 5  |-  ( `' `' H  =  H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  B ) ) )
198, 18sylbi 119 . . . 4  |-  ( Rel 
H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <->  H  Isom  R ,  S  ( A ,  B ) ) )
2017, 7, 193syl 17 . . 3  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <->  H  Isom  R ,  S  ( A ,  B ) ) )
2116, 20mpbid 145 . 2  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H  Isom  R ,  S  ( A ,  B ) )
2212, 21impbii 124 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289    i^i cin 2998    X. cxp 4436   `'ccnv 4437   Rel wrel 4443   -1-1-onto->wf1o 5014    Isom wiso 5016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024
This theorem is referenced by: (None)
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