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Theorem fival 7926
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 3054 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpr 463 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  |^| x )
3 inss1 3652 . . . . . . . . . 10  |-  ( ~P A  i^i  Fin )  C_ 
~P A
43sseli 3428 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
54elpwid 3961 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
6 eqvisset 3053 . . . . . . . . 9  |-  ( y  =  |^| x  ->  |^| x  e.  _V )
7 intex 4559 . . . . . . . . 9  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
86, 7sylibr 216 . . . . . . . 8  |-  ( y  =  |^| x  ->  x  =/=  (/) )
9 intssuni2 4260 . . . . . . . 8  |-  ( ( x  C_  A  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. A )
105, 8, 9syl2an 480 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  |^| x  C_ 
U. A )
112, 10eqsstrd 3466 . . . . . 6  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  C_ 
U. A )
12 selpw 3958 . . . . . 6  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
1311, 12sylibr 216 . . . . 5  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  e.  ~P U. A )
1413rexlimiva 2875 . . . 4  |-  ( E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x  ->  y  e.  ~P U. A )
1514abssi 3504 . . 3  |-  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  C_  ~P U. A
16 uniexg 6588 . . . 4  |-  ( A  e.  V  ->  U. A  e.  _V )
17 pwexg 4587 . . . 4  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
1816, 17syl 17 . . 3  |-  ( A  e.  V  ->  ~P U. A  e.  _V )
19 ssexg 4549 . . 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 669 . 2  |-  ( A  e.  V  ->  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  e.  _V )
21 pweq 3954 . . . . . 6  |-  ( z  =  A  ->  ~P z  =  ~P A
)
2221ineq1d 3633 . . . . 5  |-  ( z  =  A  ->  ( ~P z  i^i  Fin )  =  ( ~P A  i^i  Fin ) )
2322rexeqdv 2994 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  ( ~P z  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x
) )
2423abbidv 2569 . . 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 7925 . . 3  |-  fi  =  ( z  e.  _V  |->  { y  |  E. x  e.  ( ~P z  i^i  Fin ) y  =  |^| x }
)
2624, 25fvmptg 5946 . 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 667 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 371    = wceq 1444    e. wcel 1887   {cab 2437    =/= wne 2622   E.wrex 2738   _Vcvv 3045    i^i cin 3403    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   U.cuni 4198   |^|cint 4234   ` cfv 5582   Fincfn 7569   ficfi 7924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-fi 7925
This theorem is referenced by:  elfi  7927  fi0  7934
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