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Theorem elgiso 24888
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 6029 . . . . 5  |-  ( g  =  G  ->  (
g GrpOpHom  h )  =  ( G GrpOpHom  h ) )
2 rneq 5037 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 f1oeq2 5608 . . . . . 6  |-  ( ran  g  =  ran  G  ->  ( f : ran  g
-1-1-onto-> ran  h  <->  f : ran  G -1-1-onto-> ran  h ) )
42, 3syl 16 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g -1-1-onto-> ran  h 
<->  f : ran  G -1-1-onto-> ran  h ) )
51, 4rabeqbidv 2896 . . . 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 6030 . . . . 5  |-  ( h  =  H  ->  ( G GrpOpHom  h )  =  ( G GrpOpHom  H ) )
7 rneq 5037 . . . . . 6  |-  ( h  =  H  ->  ran  h  =  ran  H )
8 f1oeq3 5609 . . . . . 6  |-  ( ran  h  =  ran  H  ->  ( f : ran  G -1-1-onto-> ran  h  <->  f : ran  G -1-1-onto-> ran 
H ) )
97, 8syl 16 . . . . 5  |-  ( h  =  H  ->  (
f : ran  G -1-1-onto-> ran  h 
<->  f : ran  G -1-1-onto-> ran  H ) )
106, 9rabeqbidv 2896 . . . 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 21798 . . . 4  |-  GrpOpIso  =  ( g  e.  GrpOp ,  h  e.  GrpOp  |->  { f  e.  ( g GrpOpHom  h )  |  f : ran  g
-1-1-onto-> ran  h } )
12 ovex 6047 . . . . 5  |-  ( G GrpOpHom  H )  e.  _V
1312rabex 4297 . . . 4  |-  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H }  e.  _V
145, 10, 11, 13ovmpt2 6150 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G  GrpOpIso  H )  =  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran  H } )
1514eleq2d 2456 . 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 5607 . . 3  |-  ( f  =  F  ->  (
f : ran  G -1-1-onto-> ran  H  <-> 
F : ran  G -1-1-onto-> ran  H ) )
1716elrab 3037 . 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 253 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2655   ran crn 4821   -1-1-onto->wf1o 5395  (class class class)co 6022   GrpOpcgr 21624   GrpOpHom cghom 21795    GrpOpIso cgiso 21797
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  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-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-giso 21798
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