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Theorem mgmrddd 25366
Description: The range of the domain of a magma equals the domain of the domain. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
mgmrddd  |-  ( G  e.  Magma  ->  ran  dom  G  =  dom  dom  G )

Proof of Theorem mgmrddd
StepHypRef Expression
1 eqid 2283 . . . 4  |-  dom  dom  G  =  dom  dom  G
21ismgm 20987 . . 3  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( dom 
dom  G  X.  dom  dom  G ) --> dom  dom  G ) )
3 fdm 5393 . . . 4  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) --> dom  dom  G  ->  dom  G  =  ( dom 
dom  G  X.  dom  dom  G ) )
4 rneq 4904 . . . . 5  |-  ( dom 
G  =  ( dom 
dom  G  X.  dom  dom  G )  ->  ran  dom  G  =  ran  ( dom  dom  G  X.  dom  dom  G
) )
5 rnxpid 5109 . . . . 5  |-  ran  ( dom  dom  G  X.  dom  dom 
G )  =  dom  dom 
G
64, 5syl6eq 2331 . . . 4  |-  ( dom 
G  =  ( dom 
dom  G  X.  dom  dom  G )  ->  ran  dom  G  =  dom  dom  G )
73, 6syl 15 . . 3  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) --> dom  dom  G  ->  ran  dom  G  =  dom  dom 
G )
82, 7syl6bi 219 . 2  |-  ( G  e.  Magma  ->  ( G  e.  Magma  ->  ran  dom  G  =  dom  dom  G )
)
98pm2.43i 43 1  |-  ( G  e.  Magma  ->  ran  dom  G  =  dom  dom  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   Magmacmagm 20985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-mgm 20986
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