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Theorem gicer 16138
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 16122 . . . . . 6  |-  ~=ph𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
2 cnvimass 5357 . . . . . . 7  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  dom GrpIso
3 gimfn 16123 . . . . . . . 8  |- GrpIso  Fn  ( Grp  X.  Grp )
4 fndm 5680 . . . . . . . 8  |-  ( GrpIso  Fn  ( Grp  X.  Grp )  ->  dom GrpIso  =  ( Grp  X. 
Grp ) )
53, 4ax-mp 5 . . . . . . 7  |-  dom GrpIso  =  ( Grp  X.  Grp )
62, 5sseqtri 3536 . . . . . 6  |-  ( `' GrpIso  " ( _V  \  1o ) )  C_  ( Grp  X.  Grp )
71, 6eqsstri 3534 . . . . 5  |-  ~=ph𝑔  C_  ( Grp  X.  Grp )
8 relxp 5110 . . . . 5  |-  Rel  ( Grp  X.  Grp )
9 relss 5090 . . . . 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 16136 . . . 4  |-  ( x 
~=ph𝑔  y  ->  y  ~=ph𝑔  x )
1312adantl 466 . . 3  |-  ( ( T.  /\  x  ~=ph𝑔  y )  ->  y  ~=ph𝑔  x )
14 gictr 16137 . . . 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 16133 . . . . 5  |-  ( x  e.  Grp  ->  x  ~=ph𝑔  x )
17 giclcl 16134 . . . . 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 7338 . 2  |-  ( T. 
->  ~=ph𝑔 
Er  Grp )
2120trud 1388 1  |-  ~=ph𝑔 
Er  Grp
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   T. wtru 1380    e. wcel 1767   _Vcvv 3113    \ cdif 3473    C_ wss 3476   class class class wbr 4447    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002   Rel wrel 5004    Fn wfn 5583   1oc1o 7124    Er wer 7309   Grpcgrp 15730   GrpIso cgim 16119    ~=ph𝑔 cgic 16120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-1o 7131  df-er 7312  df-map 7423  df-0g 14700  df-mnd 15735  df-mhm 15789  df-grp 15871  df-ghm 16079  df-gim 16121  df-gic 16122
This theorem is referenced by: (None)
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