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Theorem submrcl 14424
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 14416 . . 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 5169 . 2  |-  dom SubMnd  C_  Mnd
3 elfvdm 5554 . 2  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  dom SubMnd )
42, 3sseldi 3178 1  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   {crab 2547   ~Pcpw 3625   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361  SubMndcsubmnd 14414
This theorem is referenced by:  submss  14427  subm0cl  14429  submcl  14430  submmnd  14431  subm0  14433  subsubm  14434  resmhm2  14437  gsumsubm  14455  gsumwsubmcl  14461  submmulgcl  14601  oppgsubm  14835  lsmub1x  14957  lsmub2x  14958  lsmsubm  14964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-submnd 14416
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