MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isores1 Unicode version

Theorem isores1 5995
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 5991 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 isores2 5994 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  <->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A ) ) ( B ,  A ) )
31, 2sylib 189 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A
) ) ( B ,  A ) )
4 isocnv 5991 . . . 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 5986 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
7 f1orel 5619 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Rel  H )
8 dfrel2 5263 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
9 isoeq1 5980 . . . . 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 188 . . . 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 19 . . 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 202 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
13 isocnv 5991 . . . . 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 204 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' H  Isom  S ,  R  ( B ,  A ) )
15 isocnv 5991 . . . 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 5986 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H : A
-1-1-onto-> B )
18 isoeq1 5980 . . . . 5  |-  ( `' `' H  =  H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  B ) ) )
198, 18sylbi 188 . . . 4  |-  ( Rel 
H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <->  H  Isom  R ,  S  ( A ,  B ) ) )
2017, 7, 193syl 19 . . 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 202 . 2  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H  Isom  R ,  S  ( A ,  B ) )
2212, 21impbii 181 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 177    = wceq 1649    i^i cin 3264    X. cxp 4818   `'ccnv 4819   Rel wrel 4825   -1-1-onto->wf1o 5395    Isom wiso 5397
This theorem is referenced by:  leiso  11637  icopnfhmeo  18841  iccpnfhmeo  18843  xrhmeo  18844  gtiso  23931  xrge0iifhmeo  24128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405
  Copyright terms: Public domain W3C validator