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Theorem gimcnv 15013
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 2408 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2408 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 14969 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frel 5557 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Rel  F )
5 dfrel2 5284 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
64, 5sylib 189 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  `' `' F  =  F )
73, 6syl 16 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  =  F )
8 id 20 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S  GrpHom  T ) )
97, 8eqeltrd 2482 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  e.  ( S  GrpHom  T ) )
109anim2i 553 . . 3  |-  ( ( `' F  e.  ( T  GrpHom  S )  /\  F  e.  ( S  GrpHom  T ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1110ancoms 440 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
12 isgim2 15011 . 2  |-  ( F  e.  ( S GrpIso  T
)  <->  ( F  e.  ( S  GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) ) )
13 isgim2 15011 . 2  |-  ( `' F  e.  ( T GrpIso  S )  <->  ( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1411, 12, 133imtr4i 258 1  |-  ( F  e.  ( S GrpIso  T
)  ->  `' F  e.  ( T GrpIso  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   `'ccnv 4840   Rel wrel 4846   -->wf 5413   ` cfv 5417  (class class class)co 6044   Basecbs 13428    GrpHom cghm 14962   GrpIso cgim 15003
This theorem is referenced by:  gicsym  15020  reloggim  20450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-mnd 14649  df-grp 14771  df-ghm 14963  df-gim 15005
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