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Theorem grpofo 25927
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 25924 . . . . 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 245 . . . 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 1020 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
51eqcomi 2460 . . 3  |-  ran  G  =  X
64, 5jctir 541 . 2  |-  ( G  e.  GrpOp  ->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
7 dffo2 5797 . 2  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
86, 7sylibr 216 1  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738    X. cxp 4832   ran crn 4835   -->wf 5578   -onto->wfo 5580  (class class class)co 6290   GrpOpcgr 25914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-ov 6293  df-grpo 25919
This theorem is referenced by:  grpocl  25928  grporndm  25938  grporn  25940  resgrprn  26008  subgores  26032  issubgoi  26038  rngosn  26132  rngodm1dm2  26146  rngosn3  26154  vcoprnelem  26197  nvgf  26237  ghomfo  30309
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