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Theorem exidcl 26566
Description: Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypothesis
Ref Expression
exidcl.1  |-  X  =  ran  G
Assertion
Ref Expression
exidcl  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem exidcl
StepHypRef Expression
1 exidcl.1 . . . . . . . 8  |-  X  =  ran  G
2 rngopid 20990 . . . . . . . 8  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
31, 2syl5eq 2327 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  X  =  dom  dom 
G )
43eleq2d 2350 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( A  e.  X  <->  A  e.  dom  dom 
G ) )
53eleq2d 2350 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( B  e.  X  <->  B  e.  dom  dom 
G ) )
64, 5anbi12d 691 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( ( A  e.  X  /\  B  e.  X )  <->  ( A  e.  dom  dom  G  /\  B  e.  dom  dom  G
) ) )
76pm5.32i 618 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( A  e.  X  /\  B  e.  X )
)  <->  ( G  e.  ( Magma  i^i  ExId  )  /\  ( A  e.  dom  dom 
G  /\  B  e.  dom  dom  G ) ) )
8 inss1 3389 . . . . . . 7  |-  ( Magma  i^i 
ExId  )  C_  Magma
98sseli 3176 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
10 eqid 2283 . . . . . . 7  |-  dom  dom  G  =  dom  dom  G
1110clmgm 20988 . . . . . 6  |-  ( ( G  e.  Magma  /\  A  e.  dom  dom  G  /\  B  e.  dom  dom  G
)  ->  ( A G B )  e.  dom  dom 
G )
129, 11syl3an1 1215 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e. 
dom  dom  G  /\  B  e.  dom  dom  G )  ->  ( A G B )  e.  dom  dom  G )
13123expb 1152 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( A  e.  dom  dom  G  /\  B  e.  dom  dom 
G ) )  -> 
( A G B )  e.  dom  dom  G )
147, 13sylbi 187 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G B )  e.  dom  dom 
G )
15143impb 1147 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e. 
dom  dom  G )
1633ad2ant1 976 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  X  =  dom  dom  G )
1715, 16eleqtrrd 2360 1  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151   dom cdm 4689   ran crn 4690  (class class class)co 5858    ExId cexid 20981   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-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-exid 20982  df-mgm 20986
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