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Theorem opidon2 23762
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 2438 . . 3  |-  dom  dom  G  =  dom  dom  G
21opidon 23760 . 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 5618 . . . 4  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  ran 
G  =  dom  dom  G )
53, 4syl5req 2483 . . 3  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  dom 
dom  G  =  X
)
6 xpeq12 4854 . . . . . . 7  |-  ( ( dom  dom  G  =  X  /\  dom  dom  G  =  X )  ->  ( dom  dom  G  X.  dom  dom 
G )  =  ( X  X.  X ) )
76anidms 645 . . . . . 6  |-  ( dom 
dom  G  =  X  ->  ( dom  dom  G  X.  dom  dom  G )  =  ( X  X.  X ) )
8 foeq2 5612 . . . . . 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 5613 . . . . 5  |-  ( dom 
dom  G  =  X  ->  ( G : ( X  X.  X )
-onto->
dom  dom  G  <->  G :
( X  X.  X
) -onto-> X ) )
119, 10bitrd 253 . . . 4  |-  ( dom 
dom  G  =  X  ->  ( G : ( dom  dom  G  X.  dom  dom  G ) -onto-> dom 
dom  G  <->  G : ( X  X.  X ) -onto-> X ) )
1211biimpd 207 . . 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 36 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    i^i cin 3322    X. cxp 4833   dom cdm 4835   ran crn 4836   -onto->wfo 5411    ExId cexid 23752   Magmacmagm 23756
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 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6089  df-exid 23753  df-mgm 23757
This theorem is referenced by:  exidreslem  28695
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