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Theorem isexid2 25454
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 rngopidOLD 25452 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
3 elin 3683 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  <-> 
( G  e.  Magma  /\  G  e.  ExId  )
)
4 eqid 2457 . . . . . . . . . . 11  |-  dom  dom  G  =  dom  dom  G
54isexid 25446 . . . . . . . . . 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 241 . . . . . . . . 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 25 . . . . . . . 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 466 . . . . . . 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 195 . . . . . 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 2472 . . . . . . 7  |-  ( ran 
G  =  dom  dom  G  ->  ( X  =  ran  G  <->  X  =  dom  dom  G ) )
11 raleq 3054 . . . . . . . 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 3063 . . . . . . 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 320 . . . . . 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 221 . . . . 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 36 . . . 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 31 . . 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 3054 . . . 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 3063 . . 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 234 . 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 5 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    i^i cin 3470   dom cdm 5008   ran crn 5009  (class class class)co 6296    ExId cexid 25443   Magmacmagm 25447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-ov 6299  df-exid 25444  df-mgmOLD 25448
This theorem is referenced by:  exidu1  25455
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