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Theorem mrisval 13845
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 5746 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3336 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  A  e.  U.
ran Moore )
4 unieq 4016 . . . . . . 7  |-  ( c  =  A  ->  U. c  =  U. A )
54pweqd 3796 . . . . . 6  |-  ( c  =  A  ->  ~P U. c  =  ~P U. A )
6 fveq2 5720 . . . . . . . . . . 11  |-  ( c  =  A  ->  (mrCls `  c )  =  (mrCls `  A ) )
7 mrisval.1 . . . . . . . . . . 11  |-  N  =  (mrCls `  A )
86, 7syl6eqr 2485 . . . . . . . . . 10  |-  ( c  =  A  ->  (mrCls `  c )  =  N )
98fveq1d 5722 . . . . . . . . 9  |-  ( c  =  A  ->  (
(mrCls `  c ) `  ( s  \  {
x } ) )  =  ( N `  ( s  \  {
x } ) ) )
109eleq2d 2502 . . . . . . . 8  |-  ( c  =  A  ->  (
x  e.  ( (mrCls `  c ) `  (
s  \  { x } ) )  <->  x  e.  ( N `  ( s 
\  { x }
) ) ) )
1110notbid 286 . . . . . . 7  |-  ( c  =  A  ->  ( -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
1211ralbidv 2717 . . . . . 6  |-  ( c  =  A  ->  ( A. x  e.  s  -.  x  e.  (
(mrCls `  c ) `  ( s  \  {
x } ) )  <->  A. x  e.  s  -.  x  e.  ( N `  ( s  \  { x } ) ) ) )
135, 12rabeqbidv 2943 . . . . 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 13803 . . . . 5  |- mrInd  =  ( c  e.  U. ran Moore  |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
) `  ( s  \  { x } ) ) } )
15 vex 2951 . . . . . . . 8  |-  c  e. 
_V
1615uniex 4697 . . . . . . 7  |-  U. c  e.  _V
1716pwex 4374 . . . . . 6  |-  ~P U. c  e.  _V
1817rabex 4346 . . . . 5  |-  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c ) `  ( s  \  {
x } ) ) }  e.  _V
1913, 14, 18fvmpt3i 5801 . . . 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 2479 . 2  |-  ( A  e.  (Moore `  X
)  ->  I  =  { s  e.  ~P U. A  |  A. x  e.  s  -.  x  e.  ( N `  (
s  \  { x } ) ) } )
22 mreuni 13815 . . . 4  |-  ( A  e.  (Moore `  X
)  ->  U. A  =  X )
2322pweqd 3796 . . 3  |-  ( A  e.  (Moore `  X
)  ->  ~P U. A  =  ~P X )
24 rabeq 2942 . . 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 2467 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 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309   ~Pcpw 3791   {csn 3806   U.cuni 4007   ran crn 4871   ` cfv 5446  Moorecmre 13797  mrClscmrc 13798  mrIndcmri 13799
This theorem is referenced by:  ismri  13846
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 13801  df-mri 13803
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