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Theorem mrisval 13548
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 5567 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3189 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  A  e.  U.
ran Moore )
4 unieq 3852 . . . . . . 7  |-  ( c  =  A  ->  U. c  =  U. A )
54pweqd 3643 . . . . . 6  |-  ( c  =  A  ->  ~P U. c  =  ~P U. A )
6 fveq2 5541 . . . . . . . . . . 11  |-  ( c  =  A  ->  (mrCls `  c )  =  (mrCls `  A ) )
7 mrisval.1 . . . . . . . . . . 11  |-  N  =  (mrCls `  A )
86, 7syl6eqr 2346 . . . . . . . . . 10  |-  ( c  =  A  ->  (mrCls `  c )  =  N )
98fveq1d 5543 . . . . . . . . 9  |-  ( c  =  A  ->  (
(mrCls `  c ) `  ( s  \  {
x } ) )  =  ( N `  ( s  \  {
x } ) ) )
109eleq2d 2363 . . . . . . . 8  |-  ( c  =  A  ->  (
x  e.  ( (mrCls `  c ) `  (
s  \  { x } ) )  <->  x  e.  ( N `  ( s 
\  { x }
) ) ) )
1110notbid 285 . . . . . . 7  |-  ( c  =  A  ->  ( -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
1211ralbidv 2576 . . . . . 6  |-  ( c  =  A  ->  ( A. x  e.  s  -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
135, 12rabeqbidv 2796 . . . . 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 13506 . . . . 5  |- mrInd  =  ( c  e.  U. ran Moore  |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
) `  ( s  \  { x } ) ) } )
15 vex 2804 . . . . . . . 8  |-  c  e. 
_V
1615uniex 4532 . . . . . . 7  |-  U. c  e.  _V
1716pwex 4209 . . . . . 6  |-  ~P U. c  e.  _V
1817rabex 4181 . . . . 5  |-  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c ) `  ( s  \  {
x } ) ) }  e.  _V
1913, 14, 18fvmpt3i 5621 . . . 4  |-  ( A  e.  U. ran Moore  ->  (mrInd `  A )  =  {
s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
203, 19syl 15 . . 3  |-  ( A  e.  (Moore `  X
)  ->  (mrInd `  A
)  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  ( s 
\  { x }
) ) } )
211, 20syl5eq 2340 . 2  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
22 mreuni 13518 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  U. A  =  X )
2322pweqd 3643 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ~P U. A  =  ~P X )
24 rabeq 2795 . . 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 15 . 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 2328 1  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162   ~Pcpw 3638   {csn 3653   U.cuni 3843   ran crn 4706   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501  mrIndcmri 13502
This theorem is referenced by:  ismri  13549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-mre 13504  df-mri 13506
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