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Theorem grporndm 20989
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 2358 . . 3  |-  ran  G  =  ran  G
21grpofo 20978 . 2  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
3 fof 5534 . . . . 5  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
4 fdm 5476 . . . . 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 4963 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  dom  G  =  dom  ( ran  G  X.  ran  G ) )
7 dmxpid 4980 . . 3  |-  dom  ( ran  G  X.  ran  G
)  =  ran  G
86, 7syl6req 2407 . 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 1642    e. wcel 1710    X. cxp 4769   dom cdm 4771   ran crn 4772   -->wf 5333   -onto->wfo 5335   GrpOpcgr 20965
This theorem is referenced by:  isabloda  21078  rngorn1  21198  vcoprne  21249  hhshsslem1  21958  divrngcl  25911  isdrngo2  25912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-ov 5948  df-grpo 20970
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