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Theorem opidon2 23983
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 2454 . . 3  |-  dom  dom  G  =  dom  dom  G
21opidon 23981 . 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 5734 . . . 4  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  ran 
G  =  dom  dom  G )
53, 4syl5req 2508 . . 3  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  dom 
dom  G  =  X
)
6 xpeq12 4970 . . . . . . 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 5728 . . . . . 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 5729 . . . . 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 1370    e. wcel 1758    i^i cin 3438    X. cxp 4949   dom cdm 4951   ran crn 4952   -onto->wfo 5527    ExId cexid 23973   Magmacmagm 23977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-ov 6206  df-exid 23974  df-mgm 23978
This theorem is referenced by:  exidreslem  28910
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