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Theorem opidon2 21754
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 2381 . . 3  |-  dom  dom  G  =  dom  dom  G
21opidon 21752 . 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 5590 . . . 4  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  ran 
G  =  dom  dom  G )
53, 4syl5req 2426 . . 3  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  dom 
dom  G  =  X
)
6 xpeq12 4831 . . . . . . 7  |-  ( ( dom  dom  G  =  X  /\  dom  dom  G  =  X )  ->  ( dom  dom  G  X.  dom  dom 
G )  =  ( X  X.  X ) )
76anidms 627 . . . . . 6  |-  ( dom 
dom  G  =  X  ->  ( dom  dom  G  X.  dom  dom  G )  =  ( X  X.  X ) )
8 foeq2 5584 . . . . . 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 16 . . . . 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 5585 . . . . 5  |-  ( dom 
dom  G  =  X  ->  ( G : ( X  X.  X )
-onto->
dom  dom  G  <->  G :
( X  X.  X
) -onto-> X ) )
119, 10bitrd 245 . . . 4  |-  ( dom 
dom  G  =  X  ->  ( G : ( dom  dom  G  X.  dom  dom  G ) -onto-> dom 
dom  G  <->  G : ( X  X.  X ) -onto-> X ) )
1211biimpd 199 . . 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 34 . 2  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  G : ( X  X.  X ) -onto-> X )
142, 13syl 16 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717    i^i cin 3256    X. cxp 4810   dom cdm 4812   ran crn 4813   -onto->wfo 5386    ExId cexid 21744   Magmacmagm 21748
This theorem is referenced by:  exidreslem  26237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pr 4338  ax-un 4635
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-fo 5394  df-fv 5396  df-ov 6017  df-exid 21745  df-mgm 21749
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