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Theorem isgim2 15791
Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 19330. (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 2441 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2441 . . 3  |-  ( Base `  S )  =  (
Base `  S )
31, 2isgim 15788 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
41, 2ghmf1o 15774 . . 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 1756   `'ccnv 4837   -1-1-onto->wf1o 5415   ` cfv 5416  (class class class)co 6089   Basecbs 14172    GrpHom cghm 15742   GrpIso cgim 15783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-mnd 15413  df-grp 15543  df-ghm 15743  df-gim 15785
This theorem is referenced by:  gimcnv  15793  gimco  15794  gicref  15797  pi1xfrgim  20628
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