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Theorem rngmgmbs4 21100
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 2669 . . . . 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 2301 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  x  =  ( u G x ) )
4 oveq2 5882 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
u G y )  =  ( u G x ) )
54eqeq2d 2307 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  =  ( u G y )  <->  x  =  ( u G x ) ) )
65rspcev 2897 . . . . . . . . 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 2667 . . . . . 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 2629 . . . . 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 6010 . . 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 5470 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    X. cxp 4703   ran crn 4706   -->wf 5267   -onto->wfo 5269  (class class class)co 5874
This theorem is referenced by:  rngorn1eq  21103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877
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