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Theorem opidon2 21007
Description: An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Hypothesis
Ref Expression
opidon2.1  |-  X  =  ran  G
Assertion
Ref Expression
opidon2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )

Proof of Theorem opidon2
StepHypRef Expression
1 eqid 2296 . . 3  |-  dom  dom  G  =  dom  dom  G
21opidon 21005 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( dom  dom  G  X.  dom  dom  G ) -onto-> dom 
dom  G )
3 opidon2.1 . . . 4  |-  X  =  ran  G
4 forn 5470 . . . 4  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  ran 
G  =  dom  dom  G )
53, 4syl5req 2341 . . 3  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  dom 
dom  G  =  X
)
6 xpeq12 4724 . . . . . . 7  |-  ( ( dom  dom  G  =  X  /\  dom  dom  G  =  X )  ->  ( dom  dom  G  X.  dom  dom 
G )  =  ( X  X.  X ) )
76anidms 626 . . . . . 6  |-  ( dom 
dom  G  =  X  ->  ( dom  dom  G  X.  dom  dom  G )  =  ( X  X.  X ) )
8 foeq2 5464 . . . . . 6  |-  ( ( dom  dom  G  X.  dom  dom  G )  =  ( X  X.  X
)  ->  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  <->  G :
( X  X.  X
) -onto-> dom  dom  G )
)
97, 8syl 15 . . . . 5  |-  ( dom 
dom  G  =  X  ->  ( G : ( dom  dom  G  X.  dom  dom  G ) -onto-> dom 
dom  G  <->  G : ( X  X.  X ) -onto-> dom 
dom  G ) )
10 foeq3 5465 . . . . 5  |-  ( dom 
dom  G  =  X  ->  ( G : ( X  X.  X )
-onto->
dom  dom  G  <->  G :
( X  X.  X
) -onto-> X ) )
119, 10bitrd 244 . . . 4  |-  ( dom 
dom  G  =  X  ->  ( G : ( dom  dom  G  X.  dom  dom  G ) -onto-> dom 
dom  G  <->  G : ( X  X.  X ) -onto-> X ) )
1211biimpd 198 . . 3  |-  ( dom 
dom  G  =  X  ->  ( G : ( dom  dom  G  X.  dom  dom  G ) -onto-> dom 
dom  G  ->  G :
( X  X.  X
) -onto-> X ) )
135, 12mpcom 32 . 2  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  G : ( X  X.  X ) -onto-> X )
142, 13syl 15 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    i^i cin 3164    X. cxp 4703   dom cdm 4705   ran crn 4706   -onto->wfo 5269    ExId cexid 20997   Magmacmagm 21001
This theorem is referenced by:  exidreslem  26670
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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