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Theorem rngmgmbs4 25210
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 2993 . . . . 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 457 . . . . . . . . 9  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
32eqcomd 2475 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  x  =  ( u G x ) )
4 oveq2 6302 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
u G y )  =  ( u G x ) )
54eqeq2d 2481 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  =  ( u G y )  <->  x  =  ( u G x ) ) )
65rspcev 3219 . . . . . . . . 9  |-  ( ( x  e.  X  /\  x  =  ( u G x ) )  ->  E. y  e.  X  x  =  ( u G y ) )
76ex 434 . . . . . . . 8  |-  ( x  e.  X  ->  (
x  =  ( u G x )  ->  E. y  e.  X  x  =  ( u G y ) ) )
83, 7syl5 32 . . . . . . 7  |-  ( x  e.  X  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  E. y  e.  X  x  =  ( u G y ) ) )
98reximdv 2941 . . . . . 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 2858 . . . . 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 569 . . 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 6443 . . 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 212 . 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 5803 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    X. cxp 5002   ran crn 5005   -->wf 5589   -onto->wfo 5591  (class class class)co 6294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-ov 6297
This theorem is referenced by:  rngorn1eq  25213
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