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Theorem submrcl 16179
Description: Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
submrcl  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )

Proof of Theorem submrcl
Dummy variables  t  x  y  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-submnd 16169 . . 3  |- SubMnd  =  ( s  e.  Mnd  |->  { t  e.  ~P ( Base `  s )  |  ( ( 0g `  s )  e.  t  /\  A. x  e.  t  A. y  e.  t  ( x ( +g  `  s ) y )  e.  t ) } )
21dmmptss 5486 . 2  |-  dom SubMnd  C_  Mnd
3 elfvdm 5874 . 2  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  dom SubMnd )
42, 3sseldi 3487 1  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804   {crab 2808   ~Pcpw 3999   dom cdm 4988   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Mndcmnd 16121  SubMndcsubmnd 16167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578  df-submnd 16169
This theorem is referenced by:  submss  16183  subm0cl  16185  submcl  16186  submmnd  16187  subm0  16189  subsubm  16190  resmhm2  16193  gsumsubm  16206  gsumwsubmcl  16208  submmulgcl  16378  oppgsubm  16599  lsmub1x  16868  lsmub2x  16869  lsmsubm  16875  submarchi  27967
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