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Theorem isexid2 21763
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 21761 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
3 elin 3475 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  <-> 
( G  e.  Magma  /\  G  e.  ExId  )
)
4 eqid 2389 . . . . . . . . . . 11  |-  dom  dom  G  =  dom  dom  G
54isexid 21755 . . . . . . . . . 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 233 . . . . . . . . 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 23 . . . . . . . 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 453 . . . . . . 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 188 . . . . . 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 2398 . . . . . . 7  |-  ( ran 
G  =  dom  dom  G  ->  ( X  =  ran  G  <->  X  =  dom  dom  G ) )
11 raleq 2849 . . . . . . . 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 2858 . . . . . . 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 312 . . . . . 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 213 . . . . 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 34 . . . 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 29 . . 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 2849 . . . 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 2858 . . 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 226 . 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 359    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652    i^i cin 3264   dom cdm 4820   ran crn 4821  (class class class)co 6022    ExId cexid 21752   Magmacmagm 21756
This theorem is referenced by:  exidu1  21764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-ov 6025  df-exid 21753  df-mgm 21757
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