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Theorem isores1 6025
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 6021 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 isores2 6024 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  <->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A ) ) ( B ,  A ) )
31, 2sylib 196 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A
) ) ( B ,  A ) )
4 isocnv 6021 . . . 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 16 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' `' H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
6 isof1o 6016 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
7 f1orel 5644 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Rel  H )
8 dfrel2 5288 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
9 isoeq1 6010 . . . . 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 195 . . . 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 20 . . 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 210 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
13 isocnv 6021 . . . . 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 212 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' H  Isom  S ,  R  ( B ,  A ) )
15 isocnv 6021 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' `' H  Isom  R ,  S  ( A ,  B ) )
1614, 15syl 16 . . 3  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' `' H  Isom  R ,  S  ( A ,  B ) )
17 isof1o 6016 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H : A
-1-1-onto-> B )
18 isoeq1 6010 . . . . 5  |-  ( `' `' H  =  H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  B ) ) )
198, 18sylbi 195 . . . 4  |-  ( Rel 
H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <->  H  Isom  R ,  S  ( A ,  B ) ) )
2017, 7, 193syl 20 . . 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 210 . 2  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H  Isom  R ,  S  ( A ,  B ) )
2212, 21impbii 188 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    i^i cin 3327    X. cxp 4838   `'ccnv 4839   Rel wrel 4845   -1-1-onto->wf1o 5417    Isom wiso 5419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427
This theorem is referenced by:  leiso  12212  icopnfhmeo  20515  iccpnfhmeo  20517  xrhmeo  20518  gtiso  25996  xrge0iifhmeo  26366
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