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Theorem isores1 5831
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 5827 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 isores2 5830 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  <->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A ) ) ( B ,  A ) )
31, 2sylib 188 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A
) ) ( B ,  A ) )
4 isocnv 5827 . . . 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 15 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' `' H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
6 isof1o 5822 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
7 f1orel 5475 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Rel  H )
8 dfrel2 5124 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
9 isoeq1 5816 . . . . 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 187 . . . 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 18 . . 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 201 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
13 isocnv 5827 . . . . 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 203 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' H  Isom  S ,  R  ( B ,  A ) )
15 isocnv 5827 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' `' H  Isom  R ,  S  ( A ,  B ) )
1614, 15syl 15 . . 3  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' `' H  Isom  R ,  S  ( A ,  B ) )
17 isof1o 5822 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H : A
-1-1-onto-> B )
18 isoeq1 5816 . . . . 5  |-  ( `' `' H  =  H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  B ) ) )
198, 18sylbi 187 . . . 4  |-  ( Rel 
H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <->  H  Isom  R ,  S  ( A ,  B ) ) )
2017, 7, 193syl 18 . . 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 201 . 2  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H  Isom  R ,  S  ( A ,  B ) )
2212, 21impbii 180 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 176    = wceq 1623    i^i cin 3151    X. cxp 4687   `'ccnv 4688   Rel wrel 4694   -1-1-onto->wf1o 5254    Isom wiso 5256
This theorem is referenced by:  leiso  11397  icopnfhmeo  18441  iccpnfhmeo  18443  xrhmeo  18444  gtiso  23241  xrge0iifhmeo  23318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264
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