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Theorem gicer 16891
Description: Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
gicer  |-  ~=g𝑔  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 16875 . . . . . 6  |-  ~=g𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
2 cnvimass 5208 . . . . . . 7  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  dom GrpIso
3 gimfn 16876 . . . . . . . 8  |- GrpIso  Fn  ( Grp  X.  Grp )
4 fndm 5693 . . . . . . . 8  |-  ( GrpIso  Fn  ( Grp  X.  Grp )  ->  dom GrpIso  =  ( Grp  X. 
Grp ) )
53, 4ax-mp 5 . . . . . . 7  |-  dom GrpIso  =  ( Grp  X.  Grp )
62, 5sseqtri 3502 . . . . . 6  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  ( Grp  X.  Grp )
71, 6eqsstri 3500 . . . . 5  |-  ~=g𝑔  C_  ( Grp 
X.  Grp )
8 relxp 4962 . . . . 5  |-  Rel  ( Grp  X.  Grp )
9 relss 4942 . . . . 5  |-  (  ~=g𝑔  C_  ( Grp  X.  Grp )  -> 
( Rel  ( Grp  X. 
Grp )  ->  Rel  ~=g𝑔  ) )
107, 8, 9mp2 9 . . . 4  |-  Rel  ~=g𝑔
1110a1i 11 . . 3  |-  ( T. 
->  Rel  ~=g𝑔  )
12 gicsym 16889 . . . 4  |-  ( x 
~=g𝑔 
y  ->  y  ~=g𝑔  x )
1312adantl 467 . . 3  |-  ( ( T.  /\  x  ~=g𝑔  y )  ->  y  ~=g𝑔  x )
14 gictr 16890 . . . 4  |-  ( ( x  ~=g𝑔  y  /\  y  ~=g𝑔  z )  ->  x  ~=g𝑔  z )
1514adantl 467 . . 3  |-  ( ( T.  /\  ( x 
~=g𝑔 
y  /\  y  ~=g𝑔  z ) )  ->  x  ~=g𝑔  z )
16 gicref 16886 . . . . 5  |-  ( x  e.  Grp  ->  x  ~=g𝑔  x )
17 giclcl 16887 . . . . 5  |-  ( x 
~=g𝑔  x  ->  x  e.  Grp )
1816, 17impbii 190 . . . 4  |-  ( x  e.  Grp  <->  x  ~=g𝑔  x )
1918a1i 11 . . 3  |-  ( T. 
->  ( x  e.  Grp  <->  x  ~=g𝑔  x ) )
2011, 13, 15, 19iserd 7397 . 2  |-  ( T. 
->  ~=g𝑔 
Er  Grp )
2120trud 1446 1  |-  ~=g𝑔  Er  Grp
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1870   _Vcvv 3087    \ cdif 3439    C_ wss 3442   class class class wbr 4426    X. cxp 4852   `'ccnv 4853   dom cdm 4854   "cima 4857   Rel wrel 4859    Fn wfn 5596   1oc1o 7183    Er wer 7368   Grpcgrp 16620   GrpIso cgim 16872    ~=g𝑔 cgic 16873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-1o 7190  df-er 7371  df-map 7482  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-grp 16624  df-ghm 16832  df-gim 16874  df-gic 16875
This theorem is referenced by: (None)
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