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Theorem exidcl 25898
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 20915 . . . . . . . 8  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
31, 2syl5eq 2300 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  X  =  dom  dom 
G )
43eleq2d 2323 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( A  e.  X  <->  A  e.  dom  dom 
G ) )
53eleq2d 2323 . . . . . 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 3331 . . . . . . 7  |-  ( Magma  i^i 
ExId  )  C_  Magma
98sseli 3118 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
10 eqid 2256 . . . . . . 7  |-  dom  dom  G  =  dom  dom  G
1110clmgm 20913 . . . . . 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 2333 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 3093   dom cdm 4626   ran crn 4627  (class class class)co 5757    ExId cexid 20906   Magmacmagm 20910
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-fo 4652  df-fv 4654  df-ov 5760  df-exid 20907  df-mgm 20911
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