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Theorem ismri 15123
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 15122 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
43eleq2d 2524 . . 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 3601 . . . . . . . 8  |-  ( s  =  S  ->  (
s  \  { x } )  =  ( S  \  { x } ) )
65fveq2d 5852 . . . . . . 7  |-  ( s  =  S  ->  ( N `  ( s  \  { x } ) )  =  ( N `
 ( S  \  { x } ) ) )
76eleq2d 2524 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( N `
 ( s  \  { x } ) )  <->  x  e.  ( N `  ( S  \  { x } ) ) ) )
87notbid 292 . . . . 5  |-  ( s  =  S  ->  ( -.  x  e.  ( N `  ( s  \  { x } ) )  <->  -.  x  e.  ( N `  ( S 
\  { x }
) ) ) )
98raleqbi1dv 3059 . . . 4  |-  ( s  =  S  ->  ( A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) )  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
109elrab 3254 . . 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 5875 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  X  e.  _V )
13 elpw2g 4600 . . . 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 702 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   ~Pcpw 3999   {csn 4016   ` cfv 5570  Moorecmre 15074  mrClscmrc 15075  mrIndcmri 15076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-mre 15078  df-mri 15080
This theorem is referenced by:  ismri2  15124  mriss  15127  lbsacsbs  18000
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