ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  submrcl Unicode version

Theorem submrcl 12739
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 12729 . 2  |- 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 ) } )
21mptrcl 5593 1  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   A.wral 2455   {crab 2459   ~Pcpw 3574   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640   Mndcmnd 12696  SubMndcsubmnd 12727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fv 5219  df-submnd 12729
This theorem is referenced by:  submss  12744  subm0cl  12746  submcl  12747  insubm  12749  submmulgcl  12901
  Copyright terms: Public domain W3C validator