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Theorem rngmgmbs4 21084
Description: The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
rngmgmbs4  |-  ( ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  ->  ran  G  =  X )
Distinct variable groups:    u, G, x    u, X, x

Proof of Theorem rngmgmbs4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.12 2656 . . . . 5  |-  ( E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  E. u  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
2 simpl 443 . . . . . . . . 9  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
32eqcomd 2288 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  x  =  ( u G x ) )
4 oveq2 5866 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
u G y )  =  ( u G x ) )
54eqeq2d 2294 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  =  ( u G y )  <->  x  =  ( u G x ) ) )
65rspcev 2884 . . . . . . . . 9  |-  ( ( x  e.  X  /\  x  =  ( u G x ) )  ->  E. y  e.  X  x  =  ( u G y ) )
76ex 423 . . . . . . . 8  |-  ( x  e.  X  ->  (
x  =  ( u G x )  ->  E. y  e.  X  x  =  ( u G y ) ) )
83, 7syl5 28 . . . . . . 7  |-  ( x  e.  X  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  E. y  e.  X  x  =  ( u G y ) ) )
98reximdv 2654 . . . . . 6  |-  ( x  e.  X  ->  ( E. u  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  E. u  e.  X  E. y  e.  X  x  =  ( u G y ) ) )
109ralimia 2616 . . . . 5  |-  ( A. x  e.  X  E. u  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  E. u  e.  X  E. y  e.  X  x  =  ( u G y ) )
111, 10syl 15 . . . 4  |-  ( E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  E. u  e.  X  E. y  e.  X  x  =  ( u G y ) )
1211anim2i 552 . . 3  |-  ( ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  -> 
( G : ( X  X.  X ) --> X  /\  A. x  e.  X  E. u  e.  X  E. y  e.  X  x  =  ( u G y ) ) )
13 foov 5994 . . 3  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  E. u  e.  X  E. y  e.  X  x  =  ( u G y ) ) )
1412, 13sylibr 203 . 2  |-  ( ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  ->  G : ( X  X.  X ) -onto-> X )
15 forn 5454 . 2  |-  ( G : ( X  X.  X ) -onto-> X  ->  ran  G  =  X )
1614, 15syl 15 1  |-  ( ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  ->  ran  G  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   ran crn 4690   -->wf 5251   -onto->wfo 5253  (class class class)co 5858
This theorem is referenced by:  rngorn1eq  21087
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-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
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