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Theorem grporn 24890
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 2467 . . . . 5  |-  ran  G  =  ran  G
32grpofo 24877 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
4 fofun 5794 . . . 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 5589 . . 3  |-  ( G  Fn  ( X  X.  X )  <->  ( Fun  G  /\  dom  G  =  ( X  X.  X
) ) )
85, 6, 7mpbir2an 918 . 2  |-  G  Fn  ( X  X.  X
)
9 fofn 5795 . . 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 5680 . . 3  |-  ( ( G  Fn  ( X  X.  X )  /\  G  Fn  ( ran  G  X.  ran  G ) )  ->  ( X  X.  X )  =  ( ran  G  X.  ran  G ) )
12 xpid11 5222 . . 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 1379    e. wcel 1767    X. cxp 4997   dom cdm 4999   ran crn 5000   Fun wfun 5580    Fn wfn 5581   -onto->wfo 5584   GrpOpcgr 24864
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-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
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-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-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-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-ov 6285  df-grpo 24869
This theorem is referenced by:  isabloi  24966  cnid  25029  addinv  25030  readdsubgo  25031  zaddsubgo  25032  mulid  25034  efghgrp  25051  cnrngo  25081  isvci  25151  cnnv  25258  cnnvba  25260  cncph  25410  hilid  25754  hhnv  25758  hhba  25760  hhph  25771  hhssnv  25856
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