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Theorem iorlid 21765
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 21764 . 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 21763 . . 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 6501 . . 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 16 . 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 2462 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   E!wreu 2652    i^i cin 3263   ran crn 4820   ` cfv 5395  (class class class)co 6021   iota_crio 6479  GIdcgi 21624    ExId cexid 21751   Magmacmagm 21755
This theorem is referenced by:  cmpidelt  21766  rngo1cl  21866
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 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-ov 6024  df-riota 6486  df-gid 21629  df-exid 21752  df-mgm 21756
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