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Theorem isgim 15901
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 15898 . . 3  |- GrpIso  =  ( a  e.  Grp , 
b  e.  Grp  |->  { c  e.  ( a 
GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
3 ovex 6218 . . . 4  |-  ( a 
GrpHom  b )  e.  _V
43rabex 4544 . . 3  |-  { c  e.  ( a  GrpHom  b )  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
5 oveq12 6202 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a  GrpHom  b )  =  ( R  GrpHom  S ) )
6 fveq2 5792 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
7 isgim.b . . . . . 6  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2510 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
9 fveq2 5792 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
10 isgim.c . . . . . 6  |-  C  =  ( Base `  S
)
119, 10syl6eqr 2510 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
12 f1oeq23 5736 . . . . 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 3066 . . 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 6410 . 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 15860 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  R  e.  Grp )
17 ghmgrp2 15861 . . . . . 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 5733 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
2221elrab 3217 . . . 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 1370    e. wcel 1758   {crab 2799   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   Basecbs 14285   Grpcgrp 15521    GrpHom cghm 15855   GrpIso cgim 15896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-ghm 15856  df-gim 15898
This theorem is referenced by:  gimf1o  15902  gimghm  15903  isgim2  15904  invoppggim  15986  lmimgim  17261  zzngim  18103  cygznlem3  18120  reefgim  22041  imasgim  29596  pmattomply1grpiso  31274
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