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Theorem mrisval 14902
Description: Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrisval.1  |-  N  =  (mrCls `  A )
mrisval.2  |-  I  =  (mrInd `  A )
Assertion
Ref Expression
mrisval  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
Distinct variable groups:    A, s, x    X, s
Allowed substitution hints:    I( x, s)    N( x, s)    X( x)

Proof of Theorem mrisval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 mrisval.2 . . 3  |-  I  =  (mrInd `  A )
2 fvssunirn 5895 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3505 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  A  e.  U.
ran Moore )
4 unieq 4259 . . . . . . 7  |-  ( c  =  A  ->  U. c  =  U. A )
54pweqd 4021 . . . . . 6  |-  ( c  =  A  ->  ~P U. c  =  ~P U. A )
6 fveq2 5872 . . . . . . . . . . 11  |-  ( c  =  A  ->  (mrCls `  c )  =  (mrCls `  A ) )
7 mrisval.1 . . . . . . . . . . 11  |-  N  =  (mrCls `  A )
86, 7syl6eqr 2526 . . . . . . . . . 10  |-  ( c  =  A  ->  (mrCls `  c )  =  N )
98fveq1d 5874 . . . . . . . . 9  |-  ( c  =  A  ->  (
(mrCls `  c ) `  ( s  \  {
x } ) )  =  ( N `  ( s  \  {
x } ) ) )
109eleq2d 2537 . . . . . . . 8  |-  ( c  =  A  ->  (
x  e.  ( (mrCls `  c ) `  (
s  \  { x } ) )  <->  x  e.  ( N `  ( s 
\  { x }
) ) ) )
1110notbid 294 . . . . . . 7  |-  ( c  =  A  ->  ( -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
1211ralbidv 2906 . . . . . 6  |-  ( c  =  A  ->  ( A. x  e.  s  -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
135, 12rabeqbidv 3113 . . . . 5  |-  ( c  =  A  ->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c ) `  ( s  \  {
x } ) ) }  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  ( s 
\  { x }
) ) } )
14 df-mri 14860 . . . . 5  |- mrInd  =  ( c  e.  U. ran Moore  |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
) `  ( s  \  { x } ) ) } )
15 vex 3121 . . . . . . . 8  |-  c  e. 
_V
1615uniex 6591 . . . . . . 7  |-  U. c  e.  _V
1716pwex 4636 . . . . . 6  |-  ~P U. c  e.  _V
1817rabex 4604 . . . . 5  |-  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c ) `  ( s  \  {
x } ) ) }  e.  _V
1913, 14, 18fvmpt3i 5961 . . . 4  |-  ( A  e.  U. ran Moore  ->  (mrInd `  A )  =  {
s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
203, 19syl 16 . . 3  |-  ( A  e.  (Moore `  X
)  ->  (mrInd `  A
)  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  ( s 
\  { x }
) ) } )
211, 20syl5eq 2520 . 2  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
22 mreuni 14872 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  U. A  =  X )
2322pweqd 4021 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ~P U. A  =  ~P X )
24 rabeq 3112 . . 3  |-  ( ~P
U. A  =  ~P X  ->  { s  e. 
~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  ( s  \  {
x } ) ) }  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) } )
2523, 24syl 16 . 2  |-  ( A  e.  (Moore `  X
)  ->  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) }  =  {
s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  ( s 
\  { x }
) ) } )
2621, 25eqtrd 2508 1  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821    \ cdif 3478   ~Pcpw 4016   {csn 4033   U.cuni 4251   ran crn 5006   ` cfv 5594  Moorecmre 14854  mrClscmrc 14855  mrIndcmri 14856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fv 5602  df-mre 14858  df-mri 14860
This theorem is referenced by:  ismri  14903
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