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Theorem gicer 14740
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 14724 . . . . . 6  |-  ~=ph𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
2 cnvimass 5033 . . . . . . 7  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  dom GrpIso
3 gimfn 14725 . . . . . . . 8  |- GrpIso  Fn  ( Grp  X.  Grp )
4 fndm 5343 . . . . . . . 8  |-  ( GrpIso  Fn  ( Grp  X.  Grp )  ->  dom GrpIso  =  ( Grp  X. 
Grp ) )
53, 4ax-mp 8 . . . . . . 7  |-  dom GrpIso  =  ( Grp  X.  Grp )
62, 5sseqtri 3210 . . . . . 6  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  ( Grp  X.  Grp )
71, 6eqsstri 3208 . . . . 5  |-  ~=ph𝑔  C_  ( Grp  X.  Grp )
8 relxp 4794 . . . . 5  |-  Rel  ( Grp  X.  Grp )
9 relss 4775 . . . . 5  |-  (  ~=ph𝑔  C_  ( Grp  X.  Grp )  -> 
( Rel  ( Grp  X. 
Grp )  ->  Rel  ~=ph𝑔  )
)
107, 8, 9mp2 17 . . . 4  |-  Rel  ~=ph𝑔
1110a1i 10 . . 3  |-  (  T. 
->  Rel  ~=ph𝑔  )
12 gicsym 14738 . . . 4  |-  ( x 
~=ph𝑔  y  ->  y  ~=ph𝑔  x )
1312adantl 452 . . 3  |-  ( (  T.  /\  x  ~=ph𝑔  y )  ->  y  ~=ph𝑔  x )
14 gictr 14739 . . . 4  |-  ( ( x  ~=ph𝑔  y  /\  y  ~=ph𝑔  z )  ->  x  ~=ph𝑔  z )
1514adantl 452 . . 3  |-  ( (  T.  /\  ( x 
~=ph𝑔  y  /\  y  ~=ph𝑔  z ) )  ->  x  ~=ph𝑔  z )
16 gicref 14735 . . . . 5  |-  ( x  e.  Grp  ->  x  ~=ph𝑔  x )
17 giclcl 14736 . . . . 5  |-  ( x 
~=ph𝑔  x  ->  x  e.  Grp )
1816, 17impbii 180 . . . 4  |-  ( x  e.  Grp  <->  x  ~=ph𝑔  x )
1918a1i 10 . . 3  |-  (  T. 
->  ( x  e.  Grp  <->  x  ~=ph𝑔  x ) )
2011, 13, 15, 19iserd 6686 . 2  |-  (  T. 
->  ~=ph𝑔 
Er  Grp )
2120trud 1314 1  |-  ~=ph𝑔 
Er  Grp
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   Rel wrel 4694    Fn wfn 5250   1oc1o 6472    Er wer 6657   Grpcgrp 14362   GrpIso cgim 14721    ~=ph𝑔 cgic 14722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-1o 6479  df-er 6660  df-map 6774  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-gim 14723  df-gic 14724
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