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Theorem mrisval 13886
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 5783 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3330 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  A  e.  U.
ran Moore )
4 unieq 4048 . . . . . . 7  |-  ( c  =  A  ->  U. c  =  U. A )
54pweqd 3828 . . . . . 6  |-  ( c  =  A  ->  ~P U. c  =  ~P U. A )
6 fveq2 5757 . . . . . . . . . . 11  |-  ( c  =  A  ->  (mrCls `  c )  =  (mrCls `  A ) )
7 mrisval.1 . . . . . . . . . . 11  |-  N  =  (mrCls `  A )
86, 7syl6eqr 2492 . . . . . . . . . 10  |-  ( c  =  A  ->  (mrCls `  c )  =  N )
98fveq1d 5759 . . . . . . . . 9  |-  ( c  =  A  ->  (
(mrCls `  c ) `  ( s  \  {
x } ) )  =  ( N `  ( s  \  {
x } ) ) )
109eleq2d 2509 . . . . . . . 8  |-  ( c  =  A  ->  (
x  e.  ( (mrCls `  c ) `  (
s  \  { x } ) )  <->  x  e.  ( N `  ( s 
\  { x }
) ) ) )
1110notbid 287 . . . . . . 7  |-  ( c  =  A  ->  ( -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
1211ralbidv 2731 . . . . . 6  |-  ( c  =  A  ->  ( A. x  e.  s  -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
135, 12rabeqbidv 2957 . . . . 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 13844 . . . . 5  |- mrInd  =  ( c  e.  U. ran Moore  |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
) `  ( s  \  { x } ) ) } )
15 vex 2965 . . . . . . . 8  |-  c  e. 
_V
1615uniex 4734 . . . . . . 7  |-  U. c  e.  _V
1716pwex 4411 . . . . . 6  |-  ~P U. c  e.  _V
1817rabex 4383 . . . . 5  |-  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c ) `  ( s  \  {
x } ) ) }  e.  _V
1913, 14, 18fvmpt3i 5838 . . . 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 2486 . 2  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
22 mreuni 13856 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  U. A  =  X )
2322pweqd 3828 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ~P U. A  =  ~P X )
24 rabeq 2956 . . 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 2474 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 1653    e. wcel 1727   A.wral 2711   {crab 2715    \ cdif 3303   ~Pcpw 3823   {csn 3838   U.cuni 4039   ran crn 4908   ` cfv 5483  Moorecmre 13838  mrClscmrc 13839  mrIndcmri 13840
This theorem is referenced by:  ismri  13887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fv 5491  df-mre 13842  df-mri 13844
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