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Theorem ismre 14848
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 5893 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  _V )
2 elex 3122 . . 3  |-  ( X  e.  C  ->  X  e.  _V )
323ad2ant2 1018 . 2  |-  ( ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  X  e.  _V )
4 pweq 4013 . . . . . . 7  |-  ( x  =  X  ->  ~P x  =  ~P X
)
54pweqd 4015 . . . . . 6  |-  ( x  =  X  ->  ~P ~P x  =  ~P ~P X )
6 eleq1 2539 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  c  <->  X  e.  c ) )
76anbi1d 704 . . . . . 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 3108 . . . . 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 14844 . . . . 5  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
10 vex 3116 . . . . . . . 8  |-  x  e. 
_V
1110pwex 4630 . . . . . . 7  |-  ~P x  e.  _V
1211pwex 4630 . . . . . 6  |-  ~P ~P x  e.  _V
1312rabex 4598 . . . . 5  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
148, 9, 13fvmpt3i 5955 . . . 4  |-  ( X  e.  _V  ->  (Moore `  X )  =  {
c  e.  ~P ~P X  |  ( X  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
1514eleq2d 2537 . . 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 2540 . . . . . 6  |-  ( c  =  C  ->  ( X  e.  c  <->  X  e.  C ) )
17 pweq 4013 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
18 eleq2 2540 . . . . . . . 8  |-  ( c  =  C  ->  ( |^| s  e.  c  <->  |^| s  e.  C ) )
1918imbi2d 316 . . . . . . 7  |-  ( c  =  C  ->  (
( s  =/=  (/)  ->  |^| s  e.  c )  <->  ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
2017, 19raleqbidv 3072 . . . . . 6  |-  ( c  =  C  ->  ( A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c )  <->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) ) )
2116, 20anbi12d 710 . . . . 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 3261 . . . 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 4631 . . . . . 6  |-  ( X  e.  _V  ->  ~P X  e.  _V )
25 elpw2g 4610 . . . . . 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 704 . . . 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 977 . . . 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 263 . . 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 279 . 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 353 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   |^|cint 4282   ` cfv 5588  Moorecmre 14840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-mre 14844
This theorem is referenced by:  mresspw  14850  mre1cl  14852  mreintcl  14853  ismred  14860
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