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Theorem exidcl 26441
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 21864 . . . . . . . 8  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
31, 2syl5eq 2448 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  X  =  dom  dom 
G )
43eleq2d 2471 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( A  e.  X  <->  A  e.  dom  dom 
G ) )
53eleq2d 2471 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( B  e.  X  <->  B  e.  dom  dom 
G ) )
64, 5anbi12d 692 . . . . 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 619 . . . 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 3521 . . . . . . 7  |-  ( Magma  i^i 
ExId  )  C_  Magma
98sseli 3304 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
10 eqid 2404 . . . . . . 7  |-  dom  dom  G  =  dom  dom  G
1110clmgm 21862 . . . . . 6  |-  ( ( G  e.  Magma  /\  A  e.  dom  dom  G  /\  B  e.  dom  dom  G
)  ->  ( A G B )  e.  dom  dom 
G )
129, 11syl3an1 1217 . . . . 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 1154 . . . 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 188 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G B )  e.  dom  dom 
G )
15143impb 1149 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e. 
dom  dom  G )
1633ad2ant1 978 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  X  =  dom  dom  G )
1715, 16eleqtrrd 2481 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3279   dom cdm 4837   ran crn 4838  (class class class)co 6040    ExId cexid 21855   Magmacmagm 21859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6043  df-exid 21856  df-mgm 21860
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