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Theorem ismndo 21321
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 21316 . . 3  |- MndOp  =  (
SemiGrp  i^i  ExId  )
21eleq2i 2430 . 2  |-  ( G  e. MndOp 
<->  G  e.  ( SemiGrp  i^i 
ExId  ) )
3 elin 3446 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
4 ismndo.1 . . . . 5  |-  X  =  dom  dom  G
54isexid 21295 . . . 4  |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
65anbi2d 684 . . 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 248 . 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 248 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 176    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629    i^i cin 3237   dom cdm 4792  (class class class)co 5981    ExId cexid 21292   SemiGrpcsem 21308  MndOpcmndo 21315
This theorem is referenced by:  ismndo1  21322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-dm 4802  df-iota 5322  df-fv 5366  df-ov 5984  df-exid 21293  df-mndo 21316
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