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Theorem rngmgmbs4 22005
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 2819 . . . . 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 2441 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  x  =  ( u G x ) )
4 oveq2 6089 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
u G y )  =  ( u G x ) )
54eqeq2d 2447 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  =  ( u G y )  <->  x  =  ( u G x ) ) )
65rspcev 3052 . . . . . . . . 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 2817 . . . . . 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 2779 . . . . 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 6220 . . 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 5656 . 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 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    X. cxp 4876   ran crn 4879   -->wf 5450   -onto->wfo 5452  (class class class)co 6081
This theorem is referenced by:  rngorn1eq  22008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084
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