MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismri Unicode version

Theorem ismri 13533
Description: Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri.1  |-  N  =  (mrCls `  A )
ismri.2  |-  I  =  (mrInd `  A )
Assertion
Ref Expression
ismri  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) ) )
Distinct variable groups:    x, A    x, S
Allowed substitution hints:    I( x)    N( x)    X( x)

Proof of Theorem ismri
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 ismri.1 . . . . 5  |-  N  =  (mrCls `  A )
2 ismri.2 . . . . 5  |-  I  =  (mrInd `  A )
31, 2mrisval 13532 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
43eleq2d 2350 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  S  e.  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) } ) )
5 difeq1 3287 . . . . . . . 8  |-  ( s  =  S  ->  (
s  \  { x } )  =  ( S  \  { x } ) )
65fveq2d 5529 . . . . . . 7  |-  ( s  =  S  ->  ( N `  ( s  \  { x } ) )  =  ( N `
 ( S  \  { x } ) ) )
76eleq2d 2350 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( N `
 ( s  \  { x } ) )  <->  x  e.  ( N `  ( S  \  { x } ) ) ) )
87notbid 285 . . . . 5  |-  ( s  =  S  ->  ( -.  x  e.  ( N `  ( s  \  { x } ) )  <->  -.  x  e.  ( N `  ( S 
\  { x }
) ) ) )
98raleqbi1dv 2744 . . . 4  |-  ( s  =  S  ->  ( A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) )  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
109elrab 2923 . . 3  |-  ( S  e.  { s  e. 
~P X  |  A. x  e.  s  -.  x  e.  ( N `  ( s  \  {
x } ) ) }  <->  ( S  e. 
~P X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) )
114, 10syl6bb 252 . 2  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  e. 
~P X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) ) )
12 elfvex 5555 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  X  e.  _V )
13 elpw2g 4174 . . . 4  |-  ( X  e.  _V  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
1412, 13syl 15 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
1514anbi1d 685 . 2  |-  ( A  e.  (Moore `  X
)  ->  ( ( S  e.  ~P X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) ) )
1611, 15bitrd 244 1  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ~Pcpw 3625   {csn 3640   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485  mrIndcmri 13486
This theorem is referenced by:  ismri2  13534  mriss  13537  lbsacsbs  15909
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  ax-un 4512
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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488  df-mri 13490
  Copyright terms: Public domain W3C validator