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Theorem ismri 14671
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 14670 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
43eleq2d 2521 . . 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 3565 . . . . . . . 8  |-  ( s  =  S  ->  (
s  \  { x } )  =  ( S  \  { x } ) )
65fveq2d 5793 . . . . . . 7  |-  ( s  =  S  ->  ( N `  ( s  \  { x } ) )  =  ( N `
 ( S  \  { x } ) ) )
76eleq2d 2521 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( N `
 ( s  \  { x } ) )  <->  x  e.  ( N `  ( S  \  { x } ) ) ) )
87notbid 294 . . . . 5  |-  ( s  =  S  ->  ( -.  x  e.  ( N `  ( s  \  { x } ) )  <->  -.  x  e.  ( N `  ( S 
\  { x }
) ) ) )
98raleqbi1dv 3021 . . . 4  |-  ( s  =  S  ->  ( A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) )  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
109elrab 3214 . . 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 261 . 2  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  e. 
~P X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) ) )
12 elfvex 5816 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  X  e.  _V )
13 elpw2g 4553 . . . 4  |-  ( X  e.  _V  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
1412, 13syl 16 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
1514anbi1d 704 . 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 253 1  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799   _Vcvv 3068    \ cdif 3423    C_ wss 3426   ~Pcpw 3958   {csn 3975   ` cfv 5516  Moorecmre 14622  mrClscmrc 14623  mrIndcmri 14624
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 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-iota 5479  df-fun 5518  df-fv 5524  df-mre 14626  df-mri 14628
This theorem is referenced by:  ismri2  14672  mriss  14675  lbsacsbs  17343
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