MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opidon Unicode version

Theorem opidon 20989
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 3389 . . . 4  |-  ( Magma  i^i 
ExId  )  C_  Magma
21sseli 3176 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
3 opidon.1 . . . . 5  |-  X  =  dom  dom  G
43ismgm 20987 . . . 4  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( X  X.  X ) --> X ) )
54ibi 232 . . 3  |-  ( G  e.  Magma  ->  G :
( X  X.  X
) --> X )
62, 5syl 15 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X ) --> X )
7 inss2 3390 . . . . 5  |-  ( Magma  i^i 
ExId  )  C_  ExId
87sseli 3176 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  ExId  )
93isexid 20984 . . . . 5  |-  ( G  e.  ExId  ->  ( G  e.  ExId  <->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
109biimpd 198 . . . 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 56 . . 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 443 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
1312ralimi 2618 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
14 oveq2 5866 . . . . . . . . . 10  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
15 id 19 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
1614, 15eqeq12d 2297 . . . . . . . . 9  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
1716rspcv 2880 . . . . . . . 8  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  -> 
( u G y )  =  y ) )
18 eqcom 2285 . . . . . . . . . . 11  |-  ( y  =  ( u G x )  <->  ( u G x )  =  y )
1914eqeq1d 2291 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( u G x )  =  y  <->  ( u G y )  =  y ) )
2018, 19syl5bb 248 . . . . . . . . . 10  |-  ( x  =  y  ->  (
y  =  ( u G x )  <->  ( u G y )  =  y ) )
2120rspcev 2884 . . . . . . . . 9  |-  ( ( y  e.  X  /\  ( u G y )  =  y )  ->  E. x  e.  X  y  =  ( u G x ) )
2221ex 423 . . . . . . . 8  |-  ( y  e.  X  ->  (
( u G y )  =  y  ->  E. x  e.  X  y  =  ( u G x ) ) )
2317, 22syld 40 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  ->  E. x  e.  X  y  =  ( u G x ) ) )
2413, 23syl5 28 . . . . . 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 2654 . . . . 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 419 . . . 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 2626 . . 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 15 . 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 5994 . 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 645 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    X. cxp 4687   dom cdm 4689   -->wf 5251   -onto->wfo 5253  (class class class)co 5858    ExId cexid 20981   Magmacmagm 20985
This theorem is referenced by:  rngopid  20990  opidon2  20991  zintdom  25438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-exid 20982  df-mgm 20986
  Copyright terms: Public domain W3C validator