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Theorem iorlid 20995
Description: A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
iorlid.1  |-  X  =  ran  G
iorlid.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
iorlid  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)

Proof of Theorem iorlid
Dummy variables  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iorlid.1 . . 3  |-  X  =  ran  G
2 iorlid.2 . . 3  |-  U  =  (GId `  G )
31, 2idrval 20994 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
41exidu1 20993 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E! u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
5 riotacl 6319 . . 3  |-  ( E! u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  e.  X )
64, 5syl 15 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  e.  X )
73, 6eqeltrd 2357 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E!wreu 2545    i^i cin 3151   ran crn 4690   ` cfv 5255  (class class class)co 5858   iota_crio 6297  GIdcgi 20854    ExId cexid 20981   Magmacmagm 20985
This theorem is referenced by:  cmpidelt  20996  rngo1cl  21096  fincmpzer  25350  expus  25365
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-reu 2550  df-rmo 2551  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-riota 6304  df-gid 20859  df-exid 20982  df-mgm 20986
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