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Theorem exidcl 29969
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 25029 . . . . . . . 8  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
31, 2syl5eq 2520 . . . . . . 7  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  X  =  dom  dom 
G )
43eleq2d 2537 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( A  e.  X  <->  A  e.  dom  dom 
G ) )
53eleq2d 2537 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( B  e.  X  <->  B  e.  dom  dom 
G ) )
64, 5anbi12d 710 . . . . 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 637 . . . 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 3718 . . . . . . 7  |-  ( Magma  i^i 
ExId  )  C_  Magma
98sseli 3500 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G  e.  Magma )
10 eqid 2467 . . . . . . 7  |-  dom  dom  G  =  dom  dom  G
1110clmgm 25027 . . . . . 6  |-  ( ( G  e.  Magma  /\  A  e.  dom  dom  G  /\  B  e.  dom  dom  G
)  ->  ( A G B )  e.  dom  dom 
G )
129, 11syl3an1 1261 . . . . 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 1197 . . . 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 195 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G B )  e.  dom  dom 
G )
15143impb 1192 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e. 
dom  dom  G )
1633ad2ant1 1017 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  X  =  dom  dom  G )
1715, 16eleqtrrd 2558 1  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475   dom cdm 4999   ran crn 5000  (class class class)co 6284    ExId cexid 25020   Magmacmagm 25024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-ov 6287  df-exid 25021  df-mgm 25025
This theorem is referenced by: (None)
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