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Theorem gicer 15804
Description: Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
gicer  |-  ~=ph𝑔 
Er  Grp

Proof of Theorem gicer
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-gic 15788 . . . . . 6  |-  ~=ph𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
2 cnvimass 5189 . . . . . . 7  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  dom GrpIso
3 gimfn 15789 . . . . . . . 8  |- GrpIso  Fn  ( Grp  X.  Grp )
4 fndm 5510 . . . . . . . 8  |-  ( GrpIso  Fn  ( Grp  X.  Grp )  ->  dom GrpIso  =  ( Grp  X. 
Grp ) )
53, 4ax-mp 5 . . . . . . 7  |-  dom GrpIso  =  ( Grp  X.  Grp )
62, 5sseqtri 3388 . . . . . 6  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  ( Grp  X.  Grp )
71, 6eqsstri 3386 . . . . 5  |-  ~=ph𝑔  C_  ( Grp  X.  Grp )
8 relxp 4947 . . . . 5  |-  Rel  ( Grp  X.  Grp )
9 relss 4927 . . . . 5  |-  (  ~=ph𝑔  C_  ( Grp  X.  Grp )  -> 
( Rel  ( Grp  X. 
Grp )  ->  Rel  ~=ph𝑔  )
)
107, 8, 9mp2 9 . . . 4  |-  Rel  ~=ph𝑔
1110a1i 11 . . 3  |-  ( T. 
->  Rel  ~=ph𝑔  )
12 gicsym 15802 . . . 4  |-  ( x 
~=ph𝑔  y  ->  y  ~=ph𝑔  x )
1312adantl 466 . . 3  |-  ( ( T.  /\  x  ~=ph𝑔  y )  ->  y  ~=ph𝑔  x )
14 gictr 15803 . . . 4  |-  ( ( x  ~=ph𝑔  y  /\  y  ~=ph𝑔  z )  ->  x  ~=ph𝑔  z )
1514adantl 466 . . 3  |-  ( ( T.  /\  ( x 
~=ph𝑔  y  /\  y  ~=ph𝑔  z ) )  ->  x  ~=ph𝑔  z )
16 gicref 15799 . . . . 5  |-  ( x  e.  Grp  ->  x  ~=ph𝑔  x )
17 giclcl 15800 . . . . 5  |-  ( x 
~=ph𝑔  x  ->  x  e.  Grp )
1816, 17impbii 188 . . . 4  |-  ( x  e.  Grp  <->  x  ~=ph𝑔  x )
1918a1i 11 . . 3  |-  ( T. 
->  ( x  e.  Grp  <->  x  ~=ph𝑔  x ) )
2011, 13, 15, 19iserd 7127 . 2  |-  ( T. 
->  ~=ph𝑔 
Er  Grp )
2120trud 1378 1  |-  ~=ph𝑔 
Er  Grp
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   T. wtru 1370    e. wcel 1756   _Vcvv 2972    \ cdif 3325    C_ wss 3328   class class class wbr 4292    X. cxp 4838   `'ccnv 4839   dom cdm 4840   "cima 4843   Rel wrel 4845    Fn wfn 5413   1oc1o 6913    Er wer 7098   Grpcgrp 15410   GrpIso cgim 15785    ~=ph𝑔 cgic 15786
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-1o 6920  df-er 7101  df-map 7216  df-0g 14380  df-mnd 15415  df-mhm 15464  df-grp 15545  df-ghm 15745  df-gim 15787  df-gic 15788
This theorem is referenced by: (None)
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