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Theorem mrisval 14589
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 5734 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3373 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  A  e.  U.
ran Moore )
4 unieq 4120 . . . . . . 7  |-  ( c  =  A  ->  U. c  =  U. A )
54pweqd 3886 . . . . . 6  |-  ( c  =  A  ->  ~P U. c  =  ~P U. A )
6 fveq2 5712 . . . . . . . . . . 11  |-  ( c  =  A  ->  (mrCls `  c )  =  (mrCls `  A ) )
7 mrisval.1 . . . . . . . . . . 11  |-  N  =  (mrCls `  A )
86, 7syl6eqr 2493 . . . . . . . . . 10  |-  ( c  =  A  ->  (mrCls `  c )  =  N )
98fveq1d 5714 . . . . . . . . 9  |-  ( c  =  A  ->  (
(mrCls `  c ) `  ( s  \  {
x } ) )  =  ( N `  ( s  \  {
x } ) ) )
109eleq2d 2510 . . . . . . . 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 2756 . . . . . 6  |-  ( c  =  A  ->  ( A. x  e.  s  -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
135, 12rabeqbidv 2988 . . . . 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 14547 . . . . 5  |- mrInd  =  ( c  e.  U. ran Moore  |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
) `  ( s  \  { x } ) ) } )
15 vex 2996 . . . . . . . 8  |-  c  e. 
_V
1615uniex 6397 . . . . . . 7  |-  U. c  e.  _V
1716pwex 4496 . . . . . 6  |-  ~P U. c  e.  _V
1817rabex 4464 . . . . 5  |-  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c ) `  ( s  \  {
x } ) ) }  e.  _V
1913, 14, 18fvmpt3i 5799 . . . 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 2487 . 2  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
22 mreuni 14559 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  U. A  =  X )
2322pweqd 3886 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ~P U. A  =  ~P X )
24 rabeq 2987 . . 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 2475 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 1369    e. wcel 1756   A.wral 2736   {crab 2740    \ cdif 3346   ~Pcpw 3881   {csn 3898   U.cuni 4112   ran crn 4862   ` cfv 5439  Moorecmre 14541  mrClscmrc 14542  mrIndcmri 14543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fv 5447  df-mre 14545  df-mri 14547
This theorem is referenced by:  ismri  14590
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