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Theorem isgim 14726
Description: An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
isgim.b  |-  B  =  ( Base `  R
)
isgim.c  |-  C  =  ( Base `  S
)
Assertion
Ref Expression
isgim  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C
) )

Proof of Theorem isgim
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-3an 936 . 2  |-  ( ( R  e.  Grp  /\  S  e.  Grp  /\  F  e.  { c  e.  ( R  GrpHom  S )  |  c : B -1-1-onto-> C }
)  <->  ( ( R  e.  Grp  /\  S  e.  Grp )  /\  F  e.  { c  e.  ( R  GrpHom  S )  |  c : B -1-1-onto-> C }
) )
2 df-gim 14723 . . 3  |- GrpIso  =  ( a  e.  Grp , 
b  e.  Grp  |->  { c  e.  ( a 
GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
3 ovex 5883 . . . 4  |-  ( a 
GrpHom  b )  e.  _V
43rabex 4165 . . 3  |-  { c  e.  ( a  GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
5 oveq12 5867 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a  GrpHom  b )  =  ( R  GrpHom  S ) )
6 fveq2 5525 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
7 isgim.b . . . . . 6  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2333 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
9 fveq2 5525 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
10 isgim.c . . . . . 6  |-  C  =  ( Base `  S
)
119, 10syl6eqr 2333 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
12 f1oeq23 5466 . . . . 5  |-  ( ( ( Base `  a
)  =  B  /\  ( Base `  b )  =  C )  ->  (
c : ( Base `  a ) -1-1-onto-> ( Base `  b
)  <->  c : B -1-1-onto-> C
) )
138, 11, 12syl2an 463 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( c : (
Base `  a ) -1-1-onto-> ( Base `  b )  <->  c : B
-1-1-onto-> C ) )
145, 13rabeqbidv 2783 . . 3  |-  ( ( a  =  R  /\  b  =  S )  ->  { c  e.  ( a  GrpHom  b )  |  c : ( Base `  a ) -1-1-onto-> ( Base `  b
) }  =  {
c  e.  ( R 
GrpHom  S )  |  c : B -1-1-onto-> C } )
152, 4, 14elovmpt2 6064 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( R  e. 
Grp  /\  S  e.  Grp  /\  F  e.  {
c  e.  ( R 
GrpHom  S )  |  c : B -1-1-onto-> C } ) )
16 ghmgrp1 14685 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  R  e.  Grp )
17 ghmgrp2 14686 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  S  e.  Grp )
1816, 17jca 518 . . . . 5  |-  ( F  e.  ( R  GrpHom  S )  ->  ( R  e.  Grp  /\  S  e. 
Grp ) )
1918adantr 451 . . . 4  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F : B -1-1-onto-> C )  ->  ( R  e.  Grp  /\  S  e.  Grp ) )
2019pm4.71ri 614 . . 3  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F : B -1-1-onto-> C )  <->  ( ( R  e.  Grp  /\  S  e.  Grp )  /\  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C ) ) )
21 f1oeq1 5463 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
2221elrab 2923 . . . 4  |-  ( F  e.  { c  e.  ( R  GrpHom  S )  |  c : B -1-1-onto-> C } 
<->  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C ) )
2322anbi2i 675 . . 3  |-  ( ( ( R  e.  Grp  /\  S  e.  Grp )  /\  F  e.  { c  e.  ( R  GrpHom  S )  |  c : B -1-1-onto-> C } )  <->  ( ( R  e.  Grp  /\  S  e.  Grp )  /\  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C ) ) )
2420, 23bitr4i 243 . 2  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F : B -1-1-onto-> C )  <->  ( ( R  e.  Grp  /\  S  e.  Grp )  /\  F  e.  { c  e.  ( R  GrpHom  S )  |  c : B -1-1-onto-> C }
) )
251, 15, 243bitr4i 268 1  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Grpcgrp 14362    GrpHom cghm 14680   GrpIso cgim 14721
This theorem is referenced by:  gimf1o  14727  gimghm  14728  isgim2  14729  invoppggim  14833  lmimgim  15818  zzngim  16506  cygznlem3  16523  reefgim  19826  imasgim  27264
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  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-ghm 14681  df-gim 14723
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