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Theorem ismndo 21931
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 21926 . . 3  |- MndOp  =  (
SemiGrp  i^i  ExId  )
21eleq2i 2500 . 2  |-  ( G  e. MndOp 
<->  G  e.  ( SemiGrp  i^i 
ExId  ) )
3 elin 3530 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
4 ismndo.1 . . . . 5  |-  X  =  dom  dom  G
54isexid 21905 . . . 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 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319   dom cdm 4878  (class class class)co 6081    ExId cexid 21902   SemiGrpcsem 21918  MndOpcmndo 21925
This theorem is referenced by:  ismndo1  21932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-dm 4888  df-iota 5418  df-fv 5462  df-ov 6084  df-exid 21903  df-mndo 21926
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