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Theorem isgim2 16101
Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 19988. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
isgim2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  `' F  e.  ( S  GrpHom  R ) ) )

Proof of Theorem isgim2
StepHypRef Expression
1 eqid 2460 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2460 . . 3  |-  ( Base `  S )  =  (
Base `  S )
31, 2isgim 16098 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
41, 2ghmf1o 16084 . . 3  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F : ( Base `  R
)
-1-1-onto-> ( Base `  S )  <->  `' F  e.  ( S 
GrpHom  R ) ) )
54pm5.32i 637 . 2  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F : ( Base `  R
)
-1-1-onto-> ( Base `  S )
)  <->  ( F  e.  ( R  GrpHom  S )  /\  `' F  e.  ( S  GrpHom  R ) ) )
63, 5bitri 249 1  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  `' F  e.  ( S  GrpHom  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1762   `'ccnv 4991   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275   Basecbs 14479    GrpHom cghm 16052   GrpIso cgim 16093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-mnd 15721  df-grp 15851  df-ghm 16053  df-gim 16095
This theorem is referenced by:  gimcnv  16103  gimco  16104  gicref  16107  pi1xfrgim  21286
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