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Theorem elgiso 24018
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 5881 . . . . 5  |-  ( g  =  G  ->  (
g GrpOpHom  h )  =  ( G GrpOpHom  h ) )
2 rneq 4920 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 f1oeq2 5480 . . . . . 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 2796 . . . 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 5882 . . . . 5  |-  ( h  =  H  ->  ( G GrpOpHom  h )  =  ( G GrpOpHom  H ) )
7 rneq 4920 . . . . . 6  |-  ( h  =  H  ->  ran  h  =  ran  H )
8 f1oeq3 5481 . . . . . 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 2796 . . . 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 21043 . . . 4  |-  GrpOpIso  =  ( g  e.  GrpOp ,  h  e.  GrpOp  |->  { f  e.  ( g GrpOpHom  h )  |  f : ran  g
-1-1-onto-> ran  h } )
12 ovex 5899 . . . . 5  |-  ( G GrpOpHom  H )  e.  _V
1312rabex 4181 . . . 4  |-  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H }  e.  _V
145, 10, 11, 13ovmpt2 5999 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G  GrpOpIso  H )  =  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran  H } )
1514eleq2d 2363 . 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 5479 . . 3  |-  ( f  =  F  ->  (
f : ran  G -1-1-onto-> ran  H  <-> 
F : ran  G -1-1-onto-> ran  H ) )
1716elrab 2936 . 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 1632    e. wcel 1696   {crab 2560   ran crn 4706   -1-1-onto->wf1o 5270  (class class class)co 5874   GrpOpcgr 20869   GrpOpHom cghom 21040    GrpOpIso cgiso 21042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-giso 21043
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