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Theorem rngmgmbs3 25417
Description: The domain of the first variable of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.)
Assertion
Ref Expression
rngmgmbs3  |-  ( ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( x G u )  =  x  /\  ( u G x )  =  x ) )  ->  dom  dom  G  =  X )
Distinct variable group:    u, X
Allowed substitution hints:    G( x, u)    X( x)

Proof of Theorem rngmgmbs3
StepHypRef Expression
1 rexn0 3556 . . 3  |-  ( E. u  e.  X  A. x  e.  X  (
( x G u )  =  x  /\  ( u G x )  =  x )  ->  X  =/=  (/) )
2 fdm 5393 . . . 4  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
3 dmeq 4879 . . . . 5  |-  ( dom 
G  =  ( X  X.  X )  ->  dom  dom  G  =  dom  ( X  X.  X
) )
4 dmxp 4897 . . . . 5  |-  ( X  =/=  (/)  ->  dom  ( X  X.  X )  =  X )
5 eqtr 2300 . . . . . 6  |-  ( ( dom  dom  G  =  dom  ( X  X.  X
)  /\  dom  ( X  X.  X )  =  X )  ->  dom  dom 
G  =  X )
65ex 423 . . . . 5  |-  ( dom 
dom  G  =  dom  ( X  X.  X
)  ->  ( dom  ( X  X.  X
)  =  X  ->  dom  dom  G  =  X ) )
73, 4, 6syl2im 34 . . . 4  |-  ( dom 
G  =  ( X  X.  X )  -> 
( X  =/=  (/)  ->  dom  dom 
G  =  X ) )
82, 7syl 15 . . 3  |-  ( G : ( X  X.  X ) --> X  -> 
( X  =/=  (/)  ->  dom  dom 
G  =  X ) )
91, 8syl5com 26 . 2  |-  ( E. u  e.  X  A. x  e.  X  (
( x G u )  =  x  /\  ( u G x )  =  x )  ->  ( G :
( X  X.  X
) --> X  ->  dom  dom 
G  =  X ) )
109impcom 419 1  |-  ( ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( x G u )  =  x  /\  ( u G x )  =  x ) )  ->  dom  dom  G  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455    X. cxp 4687   dom cdm 4689   -->wf 5251  (class class class)co 5858
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-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
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-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-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-fn 5258  df-f 5259
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