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Theorem rngmgmbs4 21958
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 2779 . . . . 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 444 . . . . . . . . 9  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
32eqcomd 2409 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  x  =  ( u G x ) )
4 oveq2 6048 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
u G y )  =  ( u G x ) )
54eqeq2d 2415 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  =  ( u G y )  <->  x  =  ( u G x ) ) )
65rspcev 3012 . . . . . . . . 9  |-  ( ( x  e.  X  /\  x  =  ( u G x ) )  ->  E. y  e.  X  x  =  ( u G y ) )
76ex 424 . . . . . . . 8  |-  ( x  e.  X  ->  (
x  =  ( u G x )  ->  E. y  e.  X  x  =  ( u G y ) ) )
83, 7syl5 30 . . . . . . 7  |-  ( x  e.  X  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  E. y  e.  X  x  =  ( u G y ) ) )
98reximdv 2777 . . . . . 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 2739 . . . . 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 16 . . . 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 553 . . 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 6179 . . 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 204 . 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 5615 . 2  |-  ( G : ( X  X.  X ) -onto-> X  ->  ran  G  =  X )
1614, 15syl 16 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 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    X. cxp 4835   ran crn 4838   -->wf 5409   -onto->wfo 5411  (class class class)co 6040
This theorem is referenced by:  rngorn1eq  21961
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6043
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