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Theorem isgim 15781
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 967 . 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 15778 . . 3  |- GrpIso  =  ( a  e.  Grp , 
b  e.  Grp  |->  { c  e.  ( a 
GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
3 ovex 6111 . . . 4  |-  ( a 
GrpHom  b )  e.  _V
43rabex 4438 . . 3  |-  { c  e.  ( a  GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
5 oveq12 6095 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a  GrpHom  b )  =  ( R  GrpHom  S ) )
6 fveq2 5686 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
7 isgim.b . . . . . 6  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2488 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
9 fveq2 5686 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
10 isgim.c . . . . . 6  |-  C  =  ( Base `  S
)
119, 10syl6eqr 2488 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
12 f1oeq23 5630 . . . . 5  |-  ( ( ( Base `  a
)  =  B  /\  ( Base `  b )  =  C )  ->  (
c : ( Base `  a ) -1-1-onto-> ( Base `  b
)  <->  c : B -1-1-onto-> C
) )
138, 11, 12syl2an 477 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( c : (
Base `  a ) -1-1-onto-> ( Base `  b )  <->  c : B
-1-1-onto-> C ) )
145, 13rabeqbidv 2962 . . 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 6302 . 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 15740 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  R  e.  Grp )
17 ghmgrp2 15741 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  S  e.  Grp )
1816, 17jca 532 . . . . 5  |-  ( F  e.  ( R  GrpHom  S )  ->  ( R  e.  Grp  /\  S  e. 
Grp ) )
1918adantr 465 . . . 4  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F : B -1-1-onto-> C )  ->  ( R  e.  Grp  /\  S  e.  Grp ) )
2019pm4.71ri 633 . . 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 5627 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
2221elrab 3112 . . . 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 694 . . 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 252 . 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 277 1  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2714   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   Basecbs 14166   Grpcgrp 15402    GrpHom cghm 15735   GrpIso cgim 15776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-ghm 15736  df-gim 15778
This theorem is referenced by:  gimf1o  15782  gimghm  15783  isgim2  15784  invoppggim  15866  lmimgim  17123  zzngim  17960  cygznlem3  17977  reefgim  21890  imasgim  29408
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