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Theorem grporndm 25783
Description: A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
Assertion
Ref Expression
grporndm  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )

Proof of Theorem grporndm
StepHypRef Expression
1 eqid 2429 . . 3  |-  ran  G  =  ran  G
21grpofo 25772 . 2  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
3 fof 5810 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
4 fdm 5750 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
53, 4syl 17 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
65dmeqd 5057 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  dom  G  =  dom  ( ran  G  X.  ran  G ) )
7 dmxpid 5074 . . 3  |-  dom  ( ran  G  X.  ran  G
)  =  ran  G
86, 7syl6req 2487 . 2  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  ran  G  =  dom  dom  G )
92, 8syl 17 1  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    X. cxp 4852   dom cdm 4854   ran crn 4855   -->wf 5597   -onto->wfo 5599   GrpOpcgr 25759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-ov 6308  df-grpo 25764
This theorem is referenced by:  isabloda  25872  rngorn1  25992  vcoprne  26043  hhshsslem1  26753  divrngcl  31903  isdrngo2  31904
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