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Theorem grporndm 21759
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 2412 . . 3  |-  ran  G  =  ran  G
21grpofo 21748 . 2  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
3 fof 5620 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
4 fdm 5562 . . . . 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 5039 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  dom  G  =  dom  ( ran  G  X.  ran  G ) )
7 dmxpid 5056 . . 3  |-  dom  ( ran  G  X.  ran  G
)  =  ran  G
86, 7syl6req 2461 . 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 1649    e. wcel 1721    X. cxp 4843   dom cdm 4845   ran crn 4846   -->wf 5417   -onto->wfo 5419   GrpOpcgr 21735
This theorem is referenced by:  isabloda  21848  rngorn1  21968  vcoprne  22019  hhshsslem1  22728  divrngcl  26471  isdrngo2  26472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429  df-ov 6051  df-grpo 21740
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