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Theorem grporndm 20877
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 2283 . . 3  |-  ran  G  =  ran  G
21grpofo 20866 . 2  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
3 fof 5451 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
4 fdm 5393 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
53, 4syl 15 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
65dmeqd 4881 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  dom  G  =  dom  ( ran  G  X.  ran  G ) )
7 dmxpid 4898 . . 3  |-  dom  ( ran  G  X.  ran  G
)  =  ran  G
86, 7syl6req 2332 . 2  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  ran  G  =  dom  dom  G )
92, 8syl 15 1  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   -onto->wfo 5253   GrpOpcgr 20853
This theorem is referenced by:  isabloda  20966  rngorn1  21086  vcoprne  21135  hhshsslem1  21844  ablocomgrp  25342  prsubrtr  25399  divrngcl  26588  isdrngo2  26589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858
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