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Theorem fival 7864
Description: The set of all the finite intersections of the elements of  A. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
fival  |-  ( A  e.  V  ->  ( fi `  A )  =  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }
)
Distinct variable groups:    x, y, A    x, V
Allowed substitution hint:    V( y)

Proof of Theorem fival
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 3115 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpr 459 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  |^| x )
3 inss1 3704 . . . . . . . . . 10  |-  ( ~P A  i^i  Fin )  C_ 
~P A
43sseli 3485 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
54elpwid 4009 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
6 eqvisset 3114 . . . . . . . . 9  |-  ( y  =  |^| x  ->  |^| x  e.  _V )
7 intex 4593 . . . . . . . . 9  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
86, 7sylibr 212 . . . . . . . 8  |-  ( y  =  |^| x  ->  x  =/=  (/) )
9 intssuni2 4297 . . . . . . . 8  |-  ( ( x  C_  A  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. A )
105, 8, 9syl2an 475 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  |^| x  C_ 
U. A )
112, 10eqsstrd 3523 . . . . . 6  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  C_ 
U. A )
12 selpw 4006 . . . . . 6  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
1311, 12sylibr 212 . . . . 5  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  e.  ~P U. A )
1413rexlimiva 2942 . . . 4  |-  ( E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x  ->  y  e.  ~P U. A )
1514abssi 3561 . . 3  |-  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  C_  ~P U. A
16 uniexg 6570 . . . 4  |-  ( A  e.  V  ->  U. A  e.  _V )
17 pwexg 4621 . . . 4  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
1816, 17syl 16 . . 3  |-  ( A  e.  V  ->  ~P U. A  e.  _V )
19 ssexg 4583 . . 3  |-  ( ( { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  C_ 
~P U. A  /\  ~P U. A  e.  _V )  ->  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  e.  _V )
2015, 18, 19sylancr 661 . 2  |-  ( A  e.  V  ->  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  e.  _V )
21 pweq 4002 . . . . . 6  |-  ( z  =  A  ->  ~P z  =  ~P A
)
2221ineq1d 3685 . . . . 5  |-  ( z  =  A  ->  ( ~P z  i^i  Fin )  =  ( ~P A  i^i  Fin ) )
2322rexeqdv 3058 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  ( ~P z  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x
) )
2423abbidv 2590 . . 3  |-  ( z  =  A  ->  { y  |  E. x  e.  ( ~P z  i^i 
Fin ) y  = 
|^| x }  =  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }
)
25 df-fi 7863 . . 3  |-  fi  =  ( z  e.  _V  |->  { y  |  E. x  e.  ( ~P z  i^i  Fin ) y  =  |^| x }
)
2624, 25fvmptg 5929 . 2  |-  ( ( A  e.  _V  /\  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  e.  _V )  ->  ( fi `  A )  =  {
y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x } )
271, 20, 26syl2anc 659 1  |-  ( A  e.  V  ->  ( fi `  A )  =  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439    =/= wne 2649   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   ` cfv 5570   Fincfn 7509   ficfi 7862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-fi 7863
This theorem is referenced by:  elfi  7865  fi0  7872
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