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Theorem grporn 20879
Description: The range of a group operation. Useful for satisfying group base set hypotheses of the form  X  =  ran  G. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grprn.1  |-  G  e. 
GrpOp
grprn.2  |-  dom  G  =  ( X  X.  X )
Assertion
Ref Expression
grporn  |-  X  =  ran  G

Proof of Theorem grporn
StepHypRef Expression
1 grprn.1 . . . 4  |-  G  e. 
GrpOp
2 eqid 2283 . . . . 5  |-  ran  G  =  ran  G
32grpofo 20866 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
4 fofun 5452 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  Fun  G )
51, 3, 4mp2b 9 . . 3  |-  Fun  G
6 grprn.2 . . 3  |-  dom  G  =  ( X  X.  X )
7 df-fn 5258 . . 3  |-  ( G  Fn  ( X  X.  X )  <->  ( Fun  G  /\  dom  G  =  ( X  X.  X
) ) )
85, 6, 7mpbir2an 886 . 2  |-  G  Fn  ( X  X.  X
)
9 fofn 5453 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G  Fn  ( ran  G  X.  ran  G ) )
101, 3, 9mp2b 9 . 2  |-  G  Fn  ( ran  G  X.  ran  G )
11 fndmu 5345 . . 3  |-  ( ( G  Fn  ( X  X.  X )  /\  G  Fn  ( ran  G  X.  ran  G ) )  ->  ( X  X.  X )  =  ( ran  G  X.  ran  G ) )
12 xpid11 4900 . . 3  |-  ( ( X  X.  X )  =  ( ran  G  X.  ran  G )  <->  X  =  ran  G )
1311, 12sylib 188 . 2  |-  ( ( G  Fn  ( X  X.  X )  /\  G  Fn  ( ran  G  X.  ran  G ) )  ->  X  =  ran  G )
148, 10, 13mp2an 653 1  |-  X  =  ran  G
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253   GrpOpcgr 20853
This theorem is referenced by:  isabloi  20955  cnid  21018  addinv  21019  readdsubgo  21020  zaddsubgo  21021  mulid  21023  efghgrp  21040  cnrngo  21070  isvci  21138  cnnv  21245  cnnvba  21247  cncph  21397  hilid  21740  hhnv  21744  hhba  21746  hhph  21757  hhssnv  21841
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-sep 4141  ax-nul 4149  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-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-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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858
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