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Theorem grpofo 23865
Description: A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1  |-  X  =  ran  G
Assertion
Ref Expression
grpofo  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )

Proof of Theorem grpofo
Dummy variables  x  y  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpfo.1 . . . . . 6  |-  X  =  ran  G
21isgrpo 23862 . . . . 5  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp 
<->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( (
x G y ) G z )  =  ( x G ( y G z ) )  /\  E. u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  E. y  e.  X  (
y G x )  =  u ) ) ) )
32ibi 241 . . . 4  |-  ( G  e.  GrpOp  ->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  (
( x G y ) G z )  =  ( x G ( y G z ) )  /\  E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) ) )
43simp1d 1000 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
51eqcomi 2467 . . 3  |-  ran  G  =  X
64, 5jctir 538 . 2  |-  ( G  e.  GrpOp  ->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
7 dffo2 5735 . 2  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
86, 7sylibr 212 1  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800    X. cxp 4949   ran crn 4952   -->wf 5525   -onto->wfo 5527  (class class class)co 6203   GrpOpcgr 23852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-ov 6206  df-grpo 23857
This theorem is referenced by:  grpocl  23866  grporndm  23876  grporn  23878  resgrprn  23946  subgores  23970  issubgoi  23976  rngosn  24070  rngodm1dm2  24084  rngosn3  24092  vcoprnelem  24135  nvgf  24175  ghomfo  27477
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