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Theorem gicref 15089
Description: Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
gicref  |-  ( R  e.  Grp  ->  R  ~=ph𝑔  R )

Proof of Theorem gicref
StepHypRef Expression
1 eqid 2442 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
21idghm 15052 . . 3  |-  ( R  e.  Grp  ->  (  _I  |`  ( Base `  R
) )  e.  ( R  GrpHom  R ) )
3 cnvresid 5552 . . . 4  |-  `' (  _I  |`  ( Base `  R ) )  =  (  _I  |`  ( Base `  R ) )
43, 2syl5eqel 2526 . . 3  |-  ( R  e.  Grp  ->  `' (  _I  |`  ( Base `  R ) )  e.  ( R  GrpHom  R ) )
5 isgim2 15083 . . 3  |-  ( (  _I  |`  ( Base `  R ) )  e.  ( R GrpIso  R )  <-> 
( (  _I  |`  ( Base `  R ) )  e.  ( R  GrpHom  R )  /\  `' (  _I  |`  ( Base `  R ) )  e.  ( R  GrpHom  R ) ) )
62, 4, 5sylanbrc 647 . 2  |-  ( R  e.  Grp  ->  (  _I  |`  ( Base `  R
) )  e.  ( R GrpIso  R ) )
7 brgici 15088 . 2  |-  ( (  _I  |`  ( Base `  R ) )  e.  ( R GrpIso  R )  ->  R  ~=ph𝑔 
R )
86, 7syl 16 1  |-  ( R  e.  Grp  ->  R  ~=ph𝑔  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1727   class class class wbr 4237    _I cid 4522   `'ccnv 4906    |` cres 4909   ` cfv 5483  (class class class)co 6110   Basecbs 13500   Grpcgrp 14716    GrpHom cghm 15034   GrpIso cgim 15075    ~=ph𝑔 cgic 15076
This theorem is referenced by:  gicer  15094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-suc 4616  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-1o 6753  df-mnd 14721  df-grp 14843  df-ghm 15035  df-gim 15077  df-gic 15078
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