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Theorem isexid2 20992
Description: If  G  e.  ( Magma  i^i  ExId  ) then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isexid2.1  |-  X  =  ran  G
Assertion
Ref Expression
isexid2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
Distinct variable groups:    u, G, x    u, X, x

Proof of Theorem isexid2
StepHypRef Expression
1 isexid2.1 . 2  |-  X  =  ran  G
2 rngopid 20990 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
3 elin 3358 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  <-> 
( G  e.  Magma  /\  G  e.  ExId  )
)
4 eqid 2283 . . . . . . . . . . 11  |-  dom  dom  G  =  dom  dom  G
54isexid 20984 . . . . . . . . . 10  |-  ( G  e.  ExId  ->  ( G  e.  ExId  <->  E. u  e.  dom  dom 
G A. x  e. 
dom  dom  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
65ibi 232 . . . . . . . . 9  |-  ( G  e.  ExId  ->  E. u  e.  dom  dom  G A. x  e.  dom  dom  G
( ( u G x )  =  x  /\  ( x G u )  =  x ) )
76a1d 22 . . . . . . . 8  |-  ( G  e.  ExId  ->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom 
G A. x  e. 
dom  dom  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
87adantl 452 . . . . . . 7  |-  ( ( G  e.  Magma  /\  G  e.  ExId  )  ->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom 
G A. x  e. 
dom  dom  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
93, 8sylbi 187 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom  G A. x  e.  dom  dom 
G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
10 eqeq2 2292 . . . . . . 7  |-  ( ran 
G  =  dom  dom  G  ->  ( X  =  ran  G  <->  X  =  dom  dom  G ) )
11 raleq 2736 . . . . . . . 8  |-  ( ran 
G  =  dom  dom  G  ->  ( A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x )  <->  A. x  e.  dom  dom  G (
( u G x )  =  x  /\  ( x G u )  =  x ) ) )
1211rexeqbi1dv 2745 . . . . . . 7  |-  ( ran 
G  =  dom  dom  G  ->  ( E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x )  <->  E. u  e.  dom  dom  G A. x  e.  dom  dom  G
( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
1310, 12imbi12d 311 . . . . . 6  |-  ( ran 
G  =  dom  dom  G  ->  ( ( X  =  ran  G  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  <->  ( X  =  dom  dom  G  ->  E. u  e.  dom  dom  G A. x  e.  dom  dom 
G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) ) )
149, 13syl5ibr 212 . . . . 5  |-  ( ran 
G  =  dom  dom  G  ->  ( G  e.  ( Magma  i^i  ExId  )  ->  ( X  =  ran  G  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) ) )
152, 14mpcom 32 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( X  =  ran  G  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
1615com12 27 . . 3  |-  ( X  =  ran  G  -> 
( G  e.  (
Magma  i^i  ExId  )  ->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
17 raleq 2736 . . . 4  |-  ( X  =  ran  G  -> 
( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
1817rexeqbi1dv 2745 . . 3  |-  ( X  =  ran  G  -> 
( E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  E. u  e.  ran  G A. x  e.  ran  G ( ( u G x )  =  x  /\  (
x G u )  =  x ) ) )
1916, 18sylibrd 225 . 2  |-  ( X  =  ran  G  -> 
( G  e.  (
Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x ) ) )
201, 19ax-mp 8 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   dom cdm 4689   ran crn 4690  (class class class)co 5858    ExId cexid 20981   Magmacmagm 20985
This theorem is referenced by:  exidu1  20993  bsmgrli  25340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-exid 20982  df-mgm 20986
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