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Theorem opidon 21903
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 3554 . . . 4  |-  ( Magma  i^i 
ExId  )  C_  Magma
21sseli 3337 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
3 opidon.1 . . . . 5  |-  X  =  dom  dom  G
43ismgm 21901 . . . 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 3555 . . . . 5  |-  ( Magma  i^i 
ExId  )  C_  ExId
87sseli 3337 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  ExId  )
93isexid 21898 . . . . 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 2774 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
14 oveq2 6082 . . . . . . . . . 10  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
15 id 20 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
1614, 15eqeq12d 2450 . . . . . . . . 9  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
1716rspcv 3041 . . . . . . . 8  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  -> 
( u G y )  =  y ) )
18 eqcom 2438 . . . . . . . . . . 11  |-  ( y  =  ( u G x )  <->  ( u G x )  =  y )
1914eqeq1d 2444 . . . . . . . . . . 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 3045 . . . . . . . . 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 2810 . . . . 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 2782 . . 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 6213 . 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 1652    e. wcel 1725   A.wral 2698   E.wrex 2699    i^i cin 3312    X. cxp 4869   dom cdm 4871   -->wf 5443   -onto->wfo 5445  (class class class)co 6074    ExId cexid 21895   Magmacmagm 21899
This theorem is referenced by:  rngopid  21904  opidon2  21905
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694
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 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fo 5453  df-fv 5455  df-ov 6077  df-exid 21896  df-mgm 21900
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