MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismre Unicode version

Theorem ismre 13492
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 5555 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  _V )
2 elex 2796 . . 3  |-  ( X  e.  C  ->  X  e.  _V )
323ad2ant2 977 . 2  |-  ( ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  X  e.  _V )
4 pweq 3628 . . . . . . 7  |-  ( x  =  X  ->  ~P x  =  ~P X
)
54pweqd 3630 . . . . . 6  |-  ( x  =  X  ->  ~P ~P x  =  ~P ~P X )
6 eleq1 2343 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  c  <->  X  e.  c ) )
76anbi1d 685 . . . . . 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 2783 . . . . 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 13488 . . . . 5  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
10 vex 2791 . . . . . . . 8  |-  x  e. 
_V
1110pwex 4193 . . . . . . 7  |-  ~P x  e.  _V
1211pwex 4193 . . . . . 6  |-  ~P ~P x  e.  _V
1312rabex 4165 . . . . 5  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
148, 9, 13fvmpt3i 5605 . . . 4  |-  ( X  e.  _V  ->  (Moore `  X )  =  {
c  e.  ~P ~P X  |  ( X  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
1514eleq2d 2350 . . 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 2344 . . . . . 6  |-  ( c  =  C  ->  ( X  e.  c  <->  X  e.  C ) )
17 pweq 3628 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
18 eleq2 2344 . . . . . . . 8  |-  ( c  =  C  ->  ( |^| s  e.  c  <->  |^| s  e.  C ) )
1918imbi2d 307 . . . . . . 7  |-  ( c  =  C  ->  (
( s  =/=  (/)  ->  |^| s  e.  c )  <->  ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
2017, 19raleqbidv 2748 . . . . . 6  |-  ( c  =  C  ->  ( A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c )  <->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) ) )
2116, 20anbi12d 691 . . . . 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 2923 . . . 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 10 . . 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 4194 . . . . . 6  |-  ( X  e.  _V  ->  ~P X  e.  _V )
25 elpw2g 4174 . . . . . 6  |-  ( ~P X  e.  _V  ->  ( C  e.  ~P ~P X 
<->  C  C_  ~P X
) )
2624, 25syl 15 . . . . 5  |-  ( X  e.  _V  ->  ( C  e.  ~P ~P X 
<->  C  C_  ~P X
) )
2726anbi1d 685 . . . 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 938 . . . 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 254 . . 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 270 . 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 342 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   ` cfv 5255  Moorecmre 13484
This theorem is referenced by:  mresspw  13494  mre1cl  13496  mreintcl  13497  ismred  13504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488
  Copyright terms: Public domain W3C validator