MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isgim Structured version   Unicode version

Theorem isgim 16427
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 973 . 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 16424 . . 3  |- GrpIso  =  ( a  e.  Grp , 
b  e.  Grp  |->  { c  e.  ( a 
GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
3 ovex 6224 . . . 4  |-  ( a 
GrpHom  b )  e.  _V
43rabex 4516 . . 3  |-  { c  e.  ( a  GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
5 oveq12 6205 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a  GrpHom  b )  =  ( R  GrpHom  S ) )
6 fveq2 5774 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
7 isgim.b . . . . . 6  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2441 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
9 fveq2 5774 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
10 isgim.c . . . . . 6  |-  C  =  ( Base `  S
)
119, 10syl6eqr 2441 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
12 f1oeq23 5718 . . . . 5  |-  ( ( ( Base `  a
)  =  B  /\  ( Base `  b )  =  C )  ->  (
c : ( Base `  a ) -1-1-onto-> ( Base `  b
)  <->  c : B -1-1-onto-> C
) )
138, 11, 12syl2an 475 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( c : (
Base `  a ) -1-1-onto-> ( Base `  b )  <->  c : B
-1-1-onto-> C ) )
145, 13rabeqbidv 3029 . . 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 6419 . 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 16386 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  R  e.  Grp )
17 ghmgrp2 16387 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  S  e.  Grp )
1816, 17jca 530 . . . . 5  |-  ( F  e.  ( R  GrpHom  S )  ->  ( R  e.  Grp  /\  S  e. 
Grp ) )
1918adantr 463 . . . 4  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F : B -1-1-onto-> C )  ->  ( R  e.  Grp  /\  S  e.  Grp ) )
2019pm4.71ri 631 . . 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 5715 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
2221elrab 3182 . . . 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 692 . . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   {crab 2736   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   Basecbs 14634   Grpcgrp 16170    GrpHom cghm 16381   GrpIso cgim 16422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-ghm 16382  df-gim 16424
This theorem is referenced by:  gimf1o  16428  gimghm  16429  isgim2  16430  invoppggim  16512  rimgim  17498  lmimgim  17824  zzngim  18682  cygznlem3  18699  pm2mpgrpiso  19403  reefgim  22930  imasgim  31216
  Copyright terms: Public domain W3C validator