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Theorem opidon2 21905
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 2436 . . 3  |-  dom  dom  G  =  dom  dom  G
21opidon 21903 . 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 5649 . . . 4  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  ran 
G  =  dom  dom  G )
53, 4syl5req 2481 . . 3  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  dom 
dom  G  =  X
)
6 xpeq12 4890 . . . . . . 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 5643 . . . . . 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 5644 . . . . 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 1652    e. wcel 1725    i^i cin 3312    X. cxp 4869   dom cdm 4871   ran crn 4872   -onto->wfo 5445    ExId cexid 21895   Magmacmagm 21899
This theorem is referenced by:  exidreslem  26544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fo 5453  df-fv 5455  df-ov 6077  df-exid 21896  df-mgm 21900
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