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Theorem ismre 14540
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 5729 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  _V )
2 elex 2993 . . 3  |-  ( X  e.  C  ->  X  e.  _V )
323ad2ant2 1010 . 2  |-  ( ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  X  e.  _V )
4 pweq 3875 . . . . . . 7  |-  ( x  =  X  ->  ~P x  =  ~P X
)
54pweqd 3877 . . . . . 6  |-  ( x  =  X  ->  ~P ~P x  =  ~P ~P X )
6 eleq1 2503 . . . . . . 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 2979 . . . . 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 14536 . . . . 5  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
10 vex 2987 . . . . . . . 8  |-  x  e. 
_V
1110pwex 4487 . . . . . . 7  |-  ~P x  e.  _V
1211pwex 4487 . . . . . 6  |-  ~P ~P x  e.  _V
1312rabex 4455 . . . . 5  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
148, 9, 13fvmpt3i 5790 . . . 4  |-  ( X  e.  _V  ->  (Moore `  X )  =  {
c  e.  ~P ~P X  |  ( X  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
1514eleq2d 2510 . . 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 2504 . . . . . 6  |-  ( c  =  C  ->  ( X  e.  c  <->  X  e.  C ) )
17 pweq 3875 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
18 eleq2 2504 . . . . . . . 8  |-  ( c  =  C  ->  ( |^| s  e.  c  <->  |^| s  e.  C ) )
1918imbi2d 316 . . . . . . 7  |-  ( c  =  C  ->  (
( s  =/=  (/)  ->  |^| s  e.  c )  <->  ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
2017, 19raleqbidv 2943 . . . . . 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 3129 . . . 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 4488 . . . . . 6  |-  ( X  e.  _V  ->  ~P X  e.  _V )
25 elpw2g 4467 . . . . . 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 969 . . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   {crab 2731   _Vcvv 2984    C_ wss 3340   (/)c0 3649   ~Pcpw 3872   |^|cint 4140   ` cfv 5430  Moorecmre 14532
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
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 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-mre 14536
This theorem is referenced by:  mresspw  14542  mre1cl  14544  mreintcl  14545  ismred  14552
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