MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpofo Unicode version

Theorem grpofo 20866
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 20863 . . . . 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 232 . . . 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 967 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
51eqcomi 2287 . . 3  |-  ran  G  =  X
64, 5jctir 524 . 2  |-  ( G  e.  GrpOp  ->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
7 dffo2 5455 . 2  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
86, 7sylibr 203 1  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   ran crn 4690   -->wf 5251   -onto->wfo 5253  (class class class)co 5858   GrpOpcgr 20853
This theorem is referenced by:  grpocl  20867  grporndm  20877  grporn  20879  resgrprn  20947  subgores  20971  issubgoi  20977  rngosn  21071  rngodm1dm2  21085  rngosn3  21093  vcoprnelem  21134  nvgf  21174  ghomfo  23998  dmrngrp  25339  ltrcmp  25405  rngodmeqrn  25419  svs2  25487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858
  Copyright terms: Public domain W3C validator