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Theorem grporn 20932
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 2316 . . . . 5  |-  ran  G  =  ran  G
32grpofo 20919 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
4 fofun 5490 . . . 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 5295 . . 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 5491 . . 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 5382 . . 3  |-  ( ( G  Fn  ( X  X.  X )  /\  G  Fn  ( ran  G  X.  ran  G ) )  ->  ( X  X.  X )  =  ( ran  G  X.  ran  G ) )
12 xpid11 4937 . . 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 1633    e. wcel 1701    X. cxp 4724   dom cdm 4726   ran crn 4727   Fun wfun 5286    Fn wfn 5287   -onto->wfo 5290   GrpOpcgr 20906
This theorem is referenced by:  isabloi  21008  cnid  21071  addinv  21072  readdsubgo  21073  zaddsubgo  21074  mulid  21076  efghgrp  21093  cnrngo  21123  isvci  21193  cnnv  21300  cnnvba  21302  cncph  21452  hilid  21795  hhnv  21799  hhba  21801  hhph  21812  hhssnv  21896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-ov 5903  df-grpo 20911
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