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Theorem opidon 23831
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 3591 . . . 4  |-  ( Magma  i^i 
ExId  )  C_  Magma
21sseli 3373 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
3 opidon.1 . . . . 5  |-  X  =  dom  dom  G
43ismgm 23829 . . . 4  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( X  X.  X ) --> X ) )
54ibi 241 . . 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 3592 . . . . 5  |-  ( Magma  i^i 
ExId  )  C_  ExId
87sseli 3373 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  ExId  )
93isexid 23826 . . . . 5  |-  ( G  e.  ExId  ->  ( G  e.  ExId  <->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
109biimpd 207 . . . 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 60 . . 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 457 . . . . . . . 8  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
1312ralimi 2812 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
14 oveq2 6120 . . . . . . . . . 10  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
15 id 22 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
1614, 15eqeq12d 2457 . . . . . . . . 9  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
1716rspcv 3090 . . . . . . . 8  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  -> 
( u G y )  =  y ) )
18 eqcom 2445 . . . . . . . . . . 11  |-  ( y  =  ( u G x )  <->  ( u G x )  =  y )
1914eqeq1d 2451 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( u G x )  =  y  <->  ( u G y )  =  y ) )
2018, 19syl5bb 257 . . . . . . . . . 10  |-  ( x  =  y  ->  (
y  =  ( u G x )  <->  ( u G y )  =  y ) )
2120rspcev 3094 . . . . . . . . 9  |-  ( ( y  e.  X  /\  ( u G y )  =  y )  ->  E. x  e.  X  y  =  ( u G x ) )
2221ex 434 . . . . . . . 8  |-  ( y  e.  X  ->  (
( u G y )  =  y  ->  E. x  e.  X  y  =  ( u G x ) ) )
2317, 22syld 44 . . . . . . 7  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  ->  E. x  e.  X  y  =  ( u G x ) ) )
2413, 23syl5 32 . . . . . 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 2848 . . . . 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 430 . . . 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 2820 . . 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 6258 . 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 664 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737    i^i cin 3348    X. cxp 4859   dom cdm 4861   -->wf 5435   -onto->wfo 5437  (class class class)co 6112    ExId cexid 23823   Magmacmagm 23827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fo 5445  df-fv 5447  df-ov 6115  df-exid 23824  df-mgm 23828
This theorem is referenced by:  rngopid  23832  opidon2  23833
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