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Theorem grpofo 21744
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 21741 . . . . 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 233 . . . 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 969 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
51eqcomi 2412 . . 3  |-  ran  G  =  X
64, 5jctir 525 . 2  |-  ( G  e.  GrpOp  ->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
7 dffo2 5620 . 2  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  ran  G  =  X ) )
86, 7sylibr 204 1  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671    X. cxp 4839   ran crn 4842   -->wf 5413   -onto->wfo 5415  (class class class)co 6044   GrpOpcgr 21731
This theorem is referenced by:  grpocl  21745  grporndm  21755  grporn  21757  resgrprn  21825  subgores  21849  issubgoi  21855  rngosn  21949  rngodm1dm2  21963  rngosn3  21971  vcoprnelem  22014  nvgf  22054  ghomfo  25059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-ov 6047  df-grpo 21736
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