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Theorem isgim 16098
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 970 . 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 16095 . . 3  |- GrpIso  =  ( a  e.  Grp , 
b  e.  Grp  |->  { c  e.  ( a 
GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
3 ovex 6300 . . . 4  |-  ( a 
GrpHom  b )  e.  _V
43rabex 4591 . . 3  |-  { c  e.  ( a  GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
5 oveq12 6284 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a  GrpHom  b )  =  ( R  GrpHom  S ) )
6 fveq2 5857 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
7 isgim.b . . . . . 6  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2519 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
9 fveq2 5857 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
10 isgim.c . . . . . 6  |-  C  =  ( Base `  S
)
119, 10syl6eqr 2519 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
12 f1oeq23 5801 . . . . 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 3101 . . 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 6495 . 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 16057 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  R  e.  Grp )
17 ghmgrp2 16058 . . . . . 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 5798 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
2221elrab 3254 . . . 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 968    = wceq 1374    e. wcel 1762   {crab 2811   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275   Basecbs 14479   Grpcgrp 15716    GrpHom cghm 16052   GrpIso cgim 16093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-ghm 16053  df-gim 16095
This theorem is referenced by:  gimf1o  16099  gimghm  16100  isgim2  16101  invoppggim  16183  rimgim  17161  lmimgim  17487  zzngim  18351  cygznlem3  18368  pm2mpgrpiso  19078  reefgim  22572  imasgim  30505
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