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Theorem elgiso 24003
Description: Membership in the set of group isomorphisms from  G to  H. (Contributed by Paul Chapman, 25-Feb-2008.)
Assertion
Ref Expression
elgiso  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) ) )

Proof of Theorem elgiso
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5865 . . . . 5  |-  ( g  =  G  ->  (
g GrpOpHom  h )  =  ( G GrpOpHom  h ) )
2 rneq 4904 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 f1oeq2 5464 . . . . . 6  |-  ( ran  g  =  ran  G  ->  ( f : ran  g
-1-1-onto-> ran  h  <->  f : ran  G -1-1-onto-> ran  h ) )
42, 3syl 15 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g -1-1-onto-> ran  h 
<->  f : ran  G -1-1-onto-> ran  h ) )
51, 4rabeqbidv 2783 . . . 4  |-  ( g  =  G  ->  { f  e.  ( g GrpOpHom  h
)  |  f : ran  g -1-1-onto-> ran  h }  =  { f  e.  ( G GrpOpHom  h )  |  f : ran  G -1-1-onto-> ran  h } )
6 oveq2 5866 . . . . 5  |-  ( h  =  H  ->  ( G GrpOpHom  h )  =  ( G GrpOpHom  H ) )
7 rneq 4904 . . . . . 6  |-  ( h  =  H  ->  ran  h  =  ran  H )
8 f1oeq3 5465 . . . . . 6  |-  ( ran  h  =  ran  H  ->  ( f : ran  G -1-1-onto-> ran  h  <->  f : ran  G -1-1-onto-> ran 
H ) )
97, 8syl 15 . . . . 5  |-  ( h  =  H  ->  (
f : ran  G -1-1-onto-> ran  h 
<->  f : ran  G -1-1-onto-> ran  H ) )
106, 9rabeqbidv 2783 . . . 4  |-  ( h  =  H  ->  { f  e.  ( G GrpOpHom  h
)  |  f : ran  G -1-1-onto-> ran  h }  =  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran  H } )
11 df-giso 21027 . . . 4  |-  GrpOpIso  =  ( g  e.  GrpOp ,  h  e.  GrpOp  |->  { f  e.  ( g GrpOpHom  h )  |  f : ran  g
-1-1-onto-> ran  h } )
12 ovex 5883 . . . . 5  |-  ( G GrpOpHom  H )  e.  _V
1312rabex 4165 . . . 4  |-  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H }  e.  _V
145, 10, 11, 13ovmpt2 5983 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G  GrpOpIso  H )  =  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran  H } )
1514eleq2d 2350 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  F  e.  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H } ) )
16 f1oeq1 5463 . . 3  |-  ( f  =  F  ->  (
f : ran  G -1-1-onto-> ran  H  <-> 
F : ran  G -1-1-onto-> ran  H ) )
1716elrab 2923 . 2  |-  ( F  e.  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H }  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) )
1815, 17syl6bb 252 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   ran crn 4690   -1-1-onto->wf1o 5254  (class class class)co 5858   GrpOpcgr 20853   GrpOpHom cghom 21024    GrpOpIso cgiso 21026
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-giso 21027
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