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Theorem gimcnv 14941
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 2366 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2366 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 14897 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frel 5498 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Rel  F )
5 dfrel2 5227 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
64, 5sylib 188 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  `' `' F  =  F )
73, 6syl 15 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  =  F )
8 id 19 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S  GrpHom  T ) )
97, 8eqeltrd 2440 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  e.  ( S  GrpHom  T ) )
109anim2i 552 . . 3  |-  ( ( `' F  e.  ( T  GrpHom  S )  /\  F  e.  ( S  GrpHom  T ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1110ancoms 439 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
12 isgim2 14939 . 2  |-  ( F  e.  ( S GrpIso  T
)  <->  ( F  e.  ( S  GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) ) )
13 isgim2 14939 . 2  |-  ( `' F  e.  ( T GrpIso  S )  <->  ( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1411, 12, 133imtr4i 257 1  |-  ( F  e.  ( S GrpIso  T
)  ->  `' F  e.  ( T GrpIso  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   `'ccnv 4791   Rel wrel 4797   -->wf 5354   ` cfv 5358  (class class class)co 5981   Basecbs 13356    GrpHom cghm 14890   GrpIso cgim 14931
This theorem is referenced by:  gicsym  14948  reloggim  20171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-mnd 14577  df-grp 14699  df-ghm 14891  df-gim 14933
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