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Theorem opidon 21758
Description: An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
opidon.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
opidon  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )

Proof of Theorem opidon
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3504 . . . 4  |-  ( Magma  i^i 
ExId  )  C_  Magma
21sseli 3287 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
3 opidon.1 . . . . 5  |-  X  =  dom  dom  G
43ismgm 21756 . . . 4  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( X  X.  X ) --> X ) )
54ibi 233 . . 3  |-  ( G  e.  Magma  ->  G :
( X  X.  X
) --> X )
62, 5syl 16 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X ) --> X )
7 inss2 3505 . . . . 5  |-  ( Magma  i^i 
ExId  )  C_  ExId
87sseli 3287 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  ExId  )
93isexid 21753 . . . . 5  |-  ( G  e.  ExId  ->  ( G  e.  ExId  <->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
109biimpd 199 . . . 4  |-  ( G  e.  ExId  ->  ( G  e.  ExId  ->  E. u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) ) )
118, 8, 10sylc 58 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
12 simpl 444 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
1312ralimi 2724 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
14 oveq2 6028 . . . . . . . . . 10  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
15 id 20 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
1614, 15eqeq12d 2401 . . . . . . . . 9  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
1716rspcv 2991 . . . . . . . 8  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  -> 
( u G y )  =  y ) )
18 eqcom 2389 . . . . . . . . . . 11  |-  ( y  =  ( u G x )  <->  ( u G x )  =  y )
1914eqeq1d 2395 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( u G x )  =  y  <->  ( u G y )  =  y ) )
2018, 19syl5bb 249 . . . . . . . . . 10  |-  ( x  =  y  ->  (
y  =  ( u G x )  <->  ( u G y )  =  y ) )
2120rspcev 2995 . . . . . . . . 9  |-  ( ( y  e.  X  /\  ( u G y )  =  y )  ->  E. x  e.  X  y  =  ( u G x ) )
2221ex 424 . . . . . . . 8  |-  ( y  e.  X  ->  (
( u G y )  =  y  ->  E. x  e.  X  y  =  ( u G x ) ) )
2317, 22syld 42 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  ->  E. x  e.  X  y  =  ( u G x ) ) )
2413, 23syl5 30 . . . . . 6  |-  ( y  e.  X  ->  ( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  E. x  e.  X  y  =  ( u G x ) ) )
2524reximdv 2760 . . . . 5  |-  ( y  e.  X  ->  ( E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  E. u  e.  X  E. x  e.  X  y  =  ( u G x ) ) )
2625impcom 420 . . . 4  |-  ( ( E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  y  e.  X )  ->  E. u  e.  X  E. x  e.  X  y  =  ( u G x ) )
2726ralrimiva 2732 . . 3  |-  ( E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. y  e.  X  E. u  e.  X  E. x  e.  X  y  =  ( u G x ) )
2811, 27syl 16 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  A. y  e.  X  E. u  e.  X  E. x  e.  X  y  =  ( u G x ) )
29 foov 6159 . 2  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  A. y  e.  X  E. u  e.  X  E. x  e.  X  y  =  ( u G x ) ) )
306, 28, 29sylanbrc 646 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    i^i cin 3262    X. cxp 4816   dom cdm 4818   -->wf 5390   -onto->wfo 5392  (class class class)co 6020    ExId cexid 21750   Magmacmagm 21754
This theorem is referenced by:  rngopid  21759  opidon2  21760
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-exid 21751  df-mgm 21755
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