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Theorem submrcl 14440
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 14432 . . 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 5185 . 2  |-  dom SubMnd  C_  Mnd
3 elfvdm 5570 . 2  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  dom SubMnd )
42, 3sseldi 3191 1  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   {crab 2560   ~Pcpw 3638   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377  SubMndcsubmnd 14430
This theorem is referenced by:  submss  14443  subm0cl  14445  submcl  14446  submmnd  14447  subm0  14449  subsubm  14450  resmhm2  14453  gsumsubm  14471  gsumwsubmcl  14477  submmulgcl  14617  oppgsubm  14851  lsmub1x  14973  lsmub2x  14974  lsmsubm  14980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-submnd 14432
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