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Theorem grporn 25628
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 2402 . . . . 5  |-  ran  G  =  ran  G
32grpofo 25615 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
4 fofun 5779 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  Fun  G )
51, 3, 4mp2b 10 . . 3  |-  Fun  G
6 grprn.2 . . 3  |-  dom  G  =  ( X  X.  X )
7 df-fn 5572 . . 3  |-  ( G  Fn  ( X  X.  X )  <->  ( Fun  G  /\  dom  G  =  ( X  X.  X
) ) )
85, 6, 7mpbir2an 921 . 2  |-  G  Fn  ( X  X.  X
)
9 fofn 5780 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G  Fn  ( ran  G  X.  ran  G ) )
101, 3, 9mp2b 10 . 2  |-  G  Fn  ( ran  G  X.  ran  G )
11 fndmu 5663 . . 3  |-  ( ( G  Fn  ( X  X.  X )  /\  G  Fn  ( ran  G  X.  ran  G ) )  ->  ( X  X.  X )  =  ( ran  G  X.  ran  G ) )
12 xpid11 5045 . . 3  |-  ( ( X  X.  X )  =  ( ran  G  X.  ran  G )  <->  X  =  ran  G )
1311, 12sylib 196 . 2  |-  ( ( G  Fn  ( X  X.  X )  /\  G  Fn  ( ran  G  X.  ran  G ) )  ->  X  =  ran  G )
148, 10, 13mp2an 670 1  |-  X  =  ran  G
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405    e. wcel 1842    X. cxp 4821   dom cdm 4823   ran crn 4824   Fun wfun 5563    Fn wfn 5564   -onto->wfo 5567   GrpOpcgr 25602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-ov 6281  df-grpo 25607
This theorem is referenced by:  isabloi  25704  cnid  25767  addinv  25768  readdsubgo  25769  zaddsubgo  25770  mulid  25772  efghgrpOLD  25789  cnrngo  25819  isvci  25889  cnnv  25996  cnnvba  25998  cncph  26148  hilid  26492  hhnv  26496  hhba  26498  hhph  26509  hhssnv  26594
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