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Theorem grpofo 25024
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 25021 . . . . 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 1008 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
51eqcomi 2480 . . 3  |-  ran  G  =  X
64, 5jctir 538 . 2  |-  ( G  e.  GrpOp  ->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
7 dffo2 5805 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    X. cxp 5003   ran crn 5006   -->wf 5590   -onto->wfo 5592  (class class class)co 6295   GrpOpcgr 25011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-ov 6298  df-grpo 25016
This theorem is referenced by:  grpocl  25025  grporndm  25035  grporn  25037  resgrprn  25105  subgores  25129  issubgoi  25135  rngosn  25229  rngodm1dm2  25243  rngosn3  25251  vcoprnelem  25294  nvgf  25334  ghomfo  28856
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