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Theorem elgiso 25099
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 6080 . . . . 5  |-  ( g  =  G  ->  (
g GrpOpHom  h )  =  ( G GrpOpHom  h ) )
2 rneq 5087 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 f1oeq2 5658 . . . . . 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 2943 . . . 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 6081 . . . . 5  |-  ( h  =  H  ->  ( G GrpOpHom  h )  =  ( G GrpOpHom  H ) )
7 rneq 5087 . . . . . 6  |-  ( h  =  H  ->  ran  h  =  ran  H )
8 f1oeq3 5659 . . . . . 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 2943 . . . 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 21940 . . . 4  |-  GrpOpIso  =  ( g  e.  GrpOp ,  h  e.  GrpOp  |->  { f  e.  ( g GrpOpHom  h )  |  f : ran  g
-1-1-onto-> ran  h } )
12 ovex 6098 . . . . 5  |-  ( G GrpOpHom  H )  e.  _V
1312rabex 4346 . . . 4  |-  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H }  e.  _V
145, 10, 11, 13ovmpt2 6201 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G  GrpOpIso  H )  =  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran  H } )
1514eleq2d 2502 . 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 5657 . . 3  |-  ( f  =  F  ->  (
f : ran  G -1-1-onto-> ran  H  <-> 
F : ran  G -1-1-onto-> ran  H ) )
1716elrab 3084 . 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 1652    e. wcel 1725   {crab 2701   ran crn 4871   -1-1-onto->wf1o 5445  (class class class)co 6073   GrpOpcgr 21766   GrpOpHom cghom 21937    GrpOpIso cgiso 21939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-giso 21940
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