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Theorem ismri 13848
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 13847 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
43eleq2d 2502 . . 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 3450 . . . . . . . 8  |-  ( s  =  S  ->  (
s  \  { x } )  =  ( S  \  { x } ) )
65fveq2d 5724 . . . . . . 7  |-  ( s  =  S  ->  ( N `  ( s  \  { x } ) )  =  ( N `
 ( S  \  { x } ) ) )
76eleq2d 2502 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( N `
 ( s  \  { x } ) )  <->  x  e.  ( N `  ( S  \  { x } ) ) ) )
87notbid 286 . . . . 5  |-  ( s  =  S  ->  ( -.  x  e.  ( N `  ( s  \  { x } ) )  <->  -.  x  e.  ( N `  ( S 
\  { x }
) ) ) )
98raleqbi1dv 2904 . . . 4  |-  ( s  =  S  ->  ( A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) )  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
109elrab 3084 . . 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 253 . 2  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  e. 
~P X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  {
x } ) ) ) ) )
12 elfvex 5750 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  X  e.  _V )
13 elpw2g 4355 . . . 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 686 . 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 245 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   ~Pcpw 3791   {csn 3806   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800  mrIndcmri 13801
This theorem is referenced by:  ismri2  13849  mriss  13852  lbsacsbs  16220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-mre 13803  df-mri 13805
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