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Theorem gimcnv 15059
Description: The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
gimcnv  |-  ( F  e.  ( S GrpIso  T
)  ->  `' F  e.  ( T GrpIso  S ) )

Proof of Theorem gimcnv
StepHypRef Expression
1 eqid 2438 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2438 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 15015 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frel 5597 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Rel  F )
5 dfrel2 5324 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
64, 5sylib 190 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  `' `' F  =  F )
73, 6syl 16 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  =  F )
8 id 21 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S  GrpHom  T ) )
97, 8eqeltrd 2512 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  e.  ( S  GrpHom  T ) )
109anim2i 554 . . 3  |-  ( ( `' F  e.  ( T  GrpHom  S )  /\  F  e.  ( S  GrpHom  T ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1110ancoms 441 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
12 isgim2 15057 . 2  |-  ( F  e.  ( S GrpIso  T
)  <->  ( F  e.  ( S  GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) ) )
13 isgim2 15057 . 2  |-  ( `' F  e.  ( T GrpIso  S )  <->  ( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1411, 12, 133imtr4i 259 1  |-  ( F  e.  ( S GrpIso  T
)  ->  `' F  e.  ( T GrpIso  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   `'ccnv 4880   Rel wrel 4886   -->wf 5453   ` cfv 5457  (class class class)co 6084   Basecbs 13474    GrpHom cghm 15008   GrpIso cgim 15049
This theorem is referenced by:  gicsym  15066  reloggim  20498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-mnd 14695  df-grp 14817  df-ghm 15009  df-gim 15051
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