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Theorem ismndo 26119
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 26114 . . 3  |- MndOp  =  (
SemiGrp  i^i  ExId  )
21eleq2i 2531 . 2  |-  ( G  e. MndOp 
<->  G  e.  ( SemiGrp  i^i 
ExId  ) )
3 elin 3628 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
4 ismndo.1 . . . . 5  |-  X  =  dom  dom  G
54isexid 26093 . . . 4  |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
65anbi2d 715 . . 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 265 . 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 265 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   E.wrex 2749    i^i cin 3414   dom cdm 4852  (class class class)co 6314    ExId cexid 26090   SemiGrpcsem 26106  MndOpcmndo 26113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-dm 4862  df-iota 5564  df-fv 5608  df-ov 6317  df-exid 26091  df-mndo 26114
This theorem is referenced by:  ismndo1  26120
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