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Theorem isores1 5808
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 5805 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 isores2 5807 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  <->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A ) ) ( B ,  A ) )
31, 2sylib 122 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  ( R  i^i  ( A  X.  A
) ) ( B ,  A ) )
4 isocnv 5805 . . . 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 5801 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
7 f1orel 5459 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Rel  H )
8 dfrel2 5074 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
9 isoeq1 5795 . . . . 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 121 . . . 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 147 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
13 isocnv 5805 . . . . 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 134 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  `' H  Isom  S ,  R  ( B ,  A ) )
15 isocnv 5805 . . . 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 5801 . . . 4  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H : A
-1-1-onto-> B )
18 isoeq1 5795 . . . . 5  |-  ( `' `' H  =  H  ->  ( `' `' H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  B ) ) )
198, 18sylbi 121 . . . 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 147 . 2  |-  ( H 
Isom  ( R  i^i  ( A  X.  A
) ) ,  S
( A ,  B
)  ->  H  Isom  R ,  S  ( A ,  B ) )
2212, 21impbii 126 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 105    = wceq 1353    i^i cin 3128    X. cxp 4620   `'ccnv 4621   Rel wrel 4627   -1-1-onto->wf1o 5210    Isom wiso 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220
This theorem is referenced by: (None)
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