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Theorem submrcl 15578
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 15569 . . 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 5434 . 2  |-  dom SubMnd  C_  Mnd
3 elfvdm 5817 . 2  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  dom SubMnd )
42, 3sseldi 3454 1  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   A.wral 2795   {crab 2799   ~Pcpw 3960   dom cdm 4940   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   0gc0g 14482   Mndcmnd 15513  SubMndcsubmnd 15567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fv 5526  df-submnd 15569
This theorem is referenced by:  submss  15582  subm0cl  15584  submcl  15585  submmnd  15586  subm0  15588  subsubm  15589  resmhm2  15592  gsumsubm  15612  gsumwsubmcl  15620  submmulgcl  15765  oppgsubm  15981  lsmub1x  16251  lsmub2x  16252  lsmsubm  16258  submarchi  26339
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