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Theorem grporndm 21829
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 2442 . . 3  |-  ran  G  =  ran  G
21grpofo 21818 . 2  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
3 fof 5682 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
4 fdm 5624 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
53, 4syl 16 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
65dmeqd 5101 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  dom  G  =  dom  ( ran  G  X.  ran  G ) )
7 dmxpid 5118 . . 3  |-  dom  ( ran  G  X.  ran  G
)  =  ran  G
86, 7syl6req 2491 . 2  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  ran  G  =  dom  dom  G )
92, 8syl 16 1  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727    X. cxp 4905   dom cdm 4907   ran crn 4908   -->wf 5479   -onto->wfo 5481   GrpOpcgr 21805
This theorem is referenced by:  isabloda  21918  rngorn1  22038  vcoprne  22089  hhshsslem1  22798  divrngcl  26611  isdrngo2  26612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fo 5489  df-fv 5491  df-ov 6113  df-grpo 21810
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