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Theorem ismre 13807
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 5750 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  _V )
2 elex 2956 . . 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 3794 . . . . . . 7  |-  ( x  =  X  ->  ~P x  =  ~P X
)
54pweqd 3796 . . . . . 6  |-  ( x  =  X  ->  ~P ~P x  =  ~P ~P X )
6 eleq1 2495 . . . . . . 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 2943 . . . . 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 13803 . . . . 5  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
10 vex 2951 . . . . . . . 8  |-  x  e. 
_V
1110pwex 4374 . . . . . . 7  |-  ~P x  e.  _V
1211pwex 4374 . . . . . 6  |-  ~P ~P x  e.  _V
1312rabex 4346 . . . . 5  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
148, 9, 13fvmpt3i 5801 . . . 4  |-  ( X  e.  _V  ->  (Moore `  X )  =  {
c  e.  ~P ~P X  |  ( X  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
1514eleq2d 2502 . . 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 2496 . . . . . 6  |-  ( c  =  C  ->  ( X  e.  c  <->  X  e.  C ) )
17 pweq 3794 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
18 eleq2 2496 . . . . . . . 8  |-  ( c  =  C  ->  ( |^| s  e.  c  <->  |^| s  e.  C ) )
1918imbi2d 308 . . . . . . 7  |-  ( c  =  C  ->  (
( s  =/=  (/)  ->  |^| s  e.  c )  <->  ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
2017, 19raleqbidv 2908 . . . . . 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 3084 . . . 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 4375 . . . . . 6  |-  ( X  e.  _V  ->  ~P X  e.  _V )
25 elpw2g 4355 . . . . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   |^|cint 4042   ` cfv 5446  Moorecmre 13799
This theorem is referenced by:  mresspw  13809  mre1cl  13811  mreintcl  13812  ismred  13819
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
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-iota 5410  df-fun 5448  df-fv 5454  df-mre 13803
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