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Theorem exidcl 25732
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 20820 . . . . . . . 8  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
31, 2syl5eq 2297 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  X  =  dom  dom 
G )
43eleq2d 2320 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( A  e.  X  <->  A  e.  dom  dom 
G ) )
53eleq2d 2320 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( B  e.  X  <->  B  e.  dom  dom 
G ) )
64, 5anbi12d 694 . . . . 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 621 . . . 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 3296 . . . . . . 7  |-  ( Magma  i^i 
ExId  )  C_  Magma
98sseli 3099 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
10 eqid 2253 . . . . . . 7  |-  dom  dom  G  =  dom  dom  G
1110clmgm 20818 . . . . . 6  |-  ( ( G  e.  Magma  /\  A  e.  dom  dom  G  /\  B  e.  dom  dom  G
)  ->  ( A G B )  e.  dom  dom 
G )
129, 11syl3an1 1220 . . . . 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 1157 . . . 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 189 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G B )  e.  dom  dom 
G )
15143impb 1152 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e. 
dom  dom  G )
1633ad2ant1 981 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  X  =  dom  dom  G )
1715, 16eleqtrrd 2330 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    i^i cin 3077   dom cdm 4580   ran crn 4581  (class class class)co 5710    ExId cexid 20811   Magmacmagm 20815
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fo 4606  df-fv 4608  df-ov 5713  df-exid 20812  df-mgm 20816
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