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Theorem ismndo 21884
Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ismndo.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
ismndo  |-  ( G  e.  A  ->  ( G  e. MndOp  <->  ( G  e.  SemiGrp 
/\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
Distinct variable groups:    x, G, y    x, X, y
Allowed substitution hints:    A( x, y)

Proof of Theorem ismndo
StepHypRef Expression
1 df-mndo 21879 . . 3  |- MndOp  =  (
SemiGrp  i^i  ExId  )
21eleq2i 2468 . 2  |-  ( G  e. MndOp 
<->  G  e.  ( SemiGrp  i^i 
ExId  ) )
3 elin 3490 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
4 ismndo.1 . . . . 5  |-  X  =  dom  dom  G
54isexid 21858 . . . 4  |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
65anbi2d 685 . . 3  |-  ( G  e.  A  ->  (
( G  e.  SemiGrp  /\  G  e.  ExId  )  <->  ( G  e.  SemiGrp  /\  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) ) )
73, 6syl5bb 249 . 2  |-  ( G  e.  A  ->  ( G  e.  ( SemiGrp  i^i 
ExId  )  <->  ( G  e.  SemiGrp 
/\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
82, 7syl5bb 249 1  |-  ( G  e.  A  ->  ( G  e. MndOp  <->  ( G  e.  SemiGrp 
/\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    i^i cin 3279   dom cdm 4837  (class class class)co 6040    ExId cexid 21855   SemiGrpcsem 21871  MndOpcmndo 21878
This theorem is referenced by:  ismndo1  21885
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-dm 4847  df-iota 5377  df-fv 5421  df-ov 6043  df-exid 21856  df-mndo 21879
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