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Theorem ismre 13742
Description: Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
ismre  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
Distinct variable groups:    C, s    X, s

Proof of Theorem ismre
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5698 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  _V )
2 elex 2907 . . 3  |-  ( X  e.  C  ->  X  e.  _V )
323ad2ant2 979 . 2  |-  ( ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  X  e.  _V )
4 pweq 3745 . . . . . . 7  |-  ( x  =  X  ->  ~P x  =  ~P X
)
54pweqd 3747 . . . . . 6  |-  ( x  =  X  ->  ~P ~P x  =  ~P ~P X )
6 eleq1 2447 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  c  <->  X  e.  c ) )
76anbi1d 686 . . . . . 6  |-  ( x  =  X  ->  (
( x  e.  c  /\  A. s  e. 
~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) )  <->  ( X  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) ) )
85, 7rabeqbidv 2894 . . . . 5  |-  ( x  =  X  ->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  =  { c  e. 
~P ~P X  | 
( X  e.  c  /\  A. s  e. 
~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
9 df-mre 13738 . . . . 5  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
10 vex 2902 . . . . . . . 8  |-  x  e. 
_V
1110pwex 4323 . . . . . . 7  |-  ~P x  e.  _V
1211pwex 4323 . . . . . 6  |-  ~P ~P x  e.  _V
1312rabex 4295 . . . . 5  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
148, 9, 13fvmpt3i 5748 . . . 4  |-  ( X  e.  _V  ->  (Moore `  X )  =  {
c  e.  ~P ~P X  |  ( X  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
1514eleq2d 2454 . . 3  |-  ( X  e.  _V  ->  ( C  e.  (Moore `  X
)  <->  C  e.  { c  e.  ~P ~P X  |  ( X  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } ) )
16 eleq2 2448 . . . . . 6  |-  ( c  =  C  ->  ( X  e.  c  <->  X  e.  C ) )
17 pweq 3745 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
18 eleq2 2448 . . . . . . . 8  |-  ( c  =  C  ->  ( |^| s  e.  c  <->  |^| s  e.  C ) )
1918imbi2d 308 . . . . . . 7  |-  ( c  =  C  ->  (
( s  =/=  (/)  ->  |^| s  e.  c )  <->  ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
2017, 19raleqbidv 2859 . . . . . 6  |-  ( c  =  C  ->  ( A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c )  <->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) ) )
2116, 20anbi12d 692 . . . . 5  |-  ( c  =  C  ->  (
( X  e.  c  /\  A. s  e. 
~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) )  <->  ( X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) ) )
2221elrab 3035 . . . 4  |-  ( C  e.  { c  e. 
~P ~P X  | 
( X  e.  c  /\  A. s  e. 
~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  <-> 
( C  e.  ~P ~P X  /\  ( X  e.  C  /\  A. s  e.  ~P  C
( s  =/=  (/)  ->  |^| s  e.  C ) ) ) )
2322a1i 11 . . 3  |-  ( X  e.  _V  ->  ( C  e.  { c  e.  ~P ~P X  | 
( X  e.  c  /\  A. s  e. 
~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  <-> 
( C  e.  ~P ~P X  /\  ( X  e.  C  /\  A. s  e.  ~P  C
( s  =/=  (/)  ->  |^| s  e.  C ) ) ) ) )
24 pwexg 4324 . . . . . 6  |-  ( X  e.  _V  ->  ~P X  e.  _V )
25 elpw2g 4304 . . . . . 6  |-  ( ~P X  e.  _V  ->  ( C  e.  ~P ~P X 
<->  C  C_  ~P X
) )
2624, 25syl 16 . . . . 5  |-  ( X  e.  _V  ->  ( C  e.  ~P ~P X 
<->  C  C_  ~P X
) )
2726anbi1d 686 . . . 4  |-  ( X  e.  _V  ->  (
( C  e.  ~P ~P X  /\  ( X  e.  C  /\  A. s  e.  ~P  C
( s  =/=  (/)  ->  |^| s  e.  C ) ) )  <-> 
( C  C_  ~P X  /\  ( X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) ) ) )
28 3anass 940 . . . 4  |-  ( ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  <->  ( C  C_ 
~P X  /\  ( X  e.  C  /\  A. s  e.  ~P  C
( s  =/=  (/)  ->  |^| s  e.  C ) ) ) )
2927, 28syl6bbr 255 . . 3  |-  ( X  e.  _V  ->  (
( C  e.  ~P ~P X  /\  ( X  e.  C  /\  A. s  e.  ~P  C
( s  =/=  (/)  ->  |^| s  e.  C ) ) )  <-> 
( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) ) ) )
3015, 23, 293bitrd 271 . 2  |-  ( X  e.  _V  ->  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) ) )
311, 3, 30pm5.21nii 343 1  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   {crab 2653   _Vcvv 2899    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   |^|cint 3992   ` cfv 5394  Moorecmre 13734
This theorem is referenced by:  mresspw  13744  mre1cl  13746  mreintcl  13747  ismred  13754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-mre 13738
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