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Theorem opidon 21005
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 3402 . . . 4  |-  ( Magma  i^i 
ExId  )  C_  Magma
21sseli 3189 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
3 opidon.1 . . . . 5  |-  X  =  dom  dom  G
43ismgm 21003 . . . 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 3403 . . . . 5  |-  ( Magma  i^i 
ExId  )  C_  ExId
87sseli 3189 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  ExId  )
93isexid 21000 . . . . 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 2631 . . . . . . 7  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
14 oveq2 5882 . . . . . . . . . 10  |-  ( x  =  y  ->  (
u G x )  =  ( u G y ) )
15 id 19 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
1614, 15eqeq12d 2310 . . . . . . . . 9  |-  ( x  =  y  ->  (
( u G x )  =  x  <->  ( u G y )  =  y ) )
1716rspcv 2893 . . . . . . . 8  |-  ( y  e.  X  ->  ( A. x  e.  X  ( u G x )  =  x  -> 
( u G y )  =  y ) )
18 eqcom 2298 . . . . . . . . . . 11  |-  ( y  =  ( u G x )  <->  ( u G x )  =  y )
1914eqeq1d 2304 . . . . . . . . . . 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 2897 . . . . . . . . 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 2667 . . . . 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 2639 . . 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 6010 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    X. cxp 4703   dom cdm 4705   -->wf 5267   -onto->wfo 5269  (class class class)co 5874    ExId cexid 20997   Magmacmagm 21001
This theorem is referenced by:  rngopid  21006  opidon2  21007  zintdom  25541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-exid 20998  df-mgm 21002
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