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Theorem grporn 23836
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 2451 . . . . 5  |-  ran  G  =  ran  G
32grpofo 23823 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
4 fofun 5721 . . . 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 5521 . . 3  |-  ( G  Fn  ( X  X.  X )  <->  ( Fun  G  /\  dom  G  =  ( X  X.  X
) ) )
85, 6, 7mpbir2an 911 . 2  |-  G  Fn  ( X  X.  X
)
9 fofn 5722 . . 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 5612 . . 3  |-  ( ( G  Fn  ( X  X.  X )  /\  G  Fn  ( ran  G  X.  ran  G ) )  ->  ( X  X.  X )  =  ( ran  G  X.  ran  G ) )
12 xpid11 5161 . . 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 672 1  |-  X  =  ran  G
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4938   dom cdm 4940   ran crn 4941   Fun wfun 5512    Fn wfn 5513   -onto->wfo 5516   GrpOpcgr 23810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526  df-ov 6195  df-grpo 23815
This theorem is referenced by:  isabloi  23912  cnid  23975  addinv  23976  readdsubgo  23977  zaddsubgo  23978  mulid  23980  efghgrp  23997  cnrngo  24027  isvci  24097  cnnv  24204  cnnvba  24206  cncph  24356  hilid  24700  hhnv  24704  hhba  24706  hhph  24717  hhssnv  24802
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