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Theorem gimcnv 16110
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 2467 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2467 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 16066 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frel 5732 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Rel  F )
5 dfrel2 5455 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
64, 5sylib 196 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  `' `' F  =  F )
73, 6syl 16 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  =  F )
8 id 22 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S  GrpHom  T ) )
97, 8eqeltrd 2555 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  `' `' F  e.  ( S  GrpHom  T ) )
109anim2i 569 . . 3  |-  ( ( `' F  e.  ( T  GrpHom  S )  /\  F  e.  ( S  GrpHom  T ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1110ancoms 453 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) )  -> 
( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
12 isgim2 16108 . 2  |-  ( F  e.  ( S GrpIso  T
)  <->  ( F  e.  ( S  GrpHom  T )  /\  `' F  e.  ( T  GrpHom  S ) ) )
13 isgim2 16108 . 2  |-  ( `' F  e.  ( T GrpIso  S )  <->  ( `' F  e.  ( T  GrpHom  S )  /\  `' `' F  e.  ( S  GrpHom  T ) ) )
1411, 12, 133imtr4i 266 1  |-  ( F  e.  ( S GrpIso  T
)  ->  `' F  e.  ( T GrpIso  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   `'ccnv 4998   Rel wrel 5004   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486    GrpHom cghm 16059   GrpIso cgim 16100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-mnd 15728  df-grp 15858  df-ghm 16060  df-gim 16102
This theorem is referenced by:  gicsym  16117  reloggim  22711  abliso  27348
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