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Theorem gicer 15063
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 15047 . . . . . 6  |-  ~=ph𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
2 cnvimass 5224 . . . . . . 7  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  dom GrpIso
3 gimfn 15048 . . . . . . . 8  |- GrpIso  Fn  ( Grp  X.  Grp )
4 fndm 5544 . . . . . . . 8  |-  ( GrpIso  Fn  ( Grp  X.  Grp )  ->  dom GrpIso  =  ( Grp  X. 
Grp ) )
53, 4ax-mp 8 . . . . . . 7  |-  dom GrpIso  =  ( Grp  X.  Grp )
62, 5sseqtri 3380 . . . . . 6  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  ( Grp  X.  Grp )
71, 6eqsstri 3378 . . . . 5  |-  ~=ph𝑔  C_  ( Grp  X.  Grp )
8 relxp 4983 . . . . 5  |-  Rel  ( Grp  X.  Grp )
9 relss 4963 . . . . 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 15061 . . . 4  |-  ( x 
~=ph𝑔  y  ->  y  ~=ph𝑔  x )
1312adantl 453 . . 3  |-  ( (  T.  /\  x  ~=ph𝑔  y )  ->  y  ~=ph𝑔  x )
14 gictr 15062 . . . 4  |-  ( ( x  ~=ph𝑔  y  /\  y  ~=ph𝑔  z )  ->  x  ~=ph𝑔  z )
1514adantl 453 . . 3  |-  ( (  T.  /\  ( x 
~=ph𝑔  y  /\  y  ~=ph𝑔  z ) )  ->  x  ~=ph𝑔  z )
16 gicref 15058 . . . . 5  |-  ( x  e.  Grp  ->  x  ~=ph𝑔  x )
17 giclcl 15059 . . . . 5  |-  ( x 
~=ph𝑔  x  ->  x  e.  Grp )
1816, 17impbii 181 . . . 4  |-  ( x  e.  Grp  <->  x  ~=ph𝑔  x )
1918a1i 11 . . 3  |-  (  T. 
->  ( x  e.  Grp  <->  x  ~=ph𝑔  x ) )
2011, 13, 15, 19iserd 6931 . 2  |-  (  T. 
->  ~=ph𝑔 
Er  Grp )
2120trud 1332 1  |-  ~=ph𝑔 
Er  Grp
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    C_ wss 3320   class class class wbr 4212    X. cxp 4876   `'ccnv 4877   dom cdm 4878   "cima 4881   Rel wrel 4883    Fn wfn 5449   1oc1o 6717    Er wer 6902   Grpcgrp 14685   GrpIso cgim 15044    ~=ph𝑔 cgic 15045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-1o 6724  df-er 6905  df-map 7020  df-0g 13727  df-mnd 14690  df-mhm 14738  df-grp 14812  df-ghm 15004  df-gim 15046  df-gic 15047
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