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Theorem fival 7932
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 3096 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpr 462 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  |^| x )
3 inss1 3688 . . . . . . . . . 10  |-  ( ~P A  i^i  Fin )  C_ 
~P A
43sseli 3466 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
54elpwid 3995 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
6 eqvisset 3095 . . . . . . . . 9  |-  ( y  =  |^| x  ->  |^| x  e.  _V )
7 intex 4581 . . . . . . . . 9  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
86, 7sylibr 215 . . . . . . . 8  |-  ( y  =  |^| x  ->  x  =/=  (/) )
9 intssuni2 4284 . . . . . . . 8  |-  ( ( x  C_  A  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. A )
105, 8, 9syl2an 479 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  |^| x  C_ 
U. A )
112, 10eqsstrd 3504 . . . . . 6  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  C_ 
U. A )
12 selpw 3992 . . . . . 6  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
1311, 12sylibr 215 . . . . 5  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  e.  ~P U. A )
1413rexlimiva 2920 . . . 4  |-  ( E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x  ->  y  e.  ~P U. A )
1514abssi 3542 . . 3  |-  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  C_  ~P U. A
16 uniexg 6602 . . . 4  |-  ( A  e.  V  ->  U. A  e.  _V )
17 pwexg 4609 . . . 4  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
1816, 17syl 17 . . 3  |-  ( A  e.  V  ->  ~P U. A  e.  _V )
19 ssexg 4571 . . 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 667 . 2  |-  ( A  e.  V  ->  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  e.  _V )
21 pweq 3988 . . . . . 6  |-  ( z  =  A  ->  ~P z  =  ~P A
)
2221ineq1d 3669 . . . . 5  |-  ( z  =  A  ->  ( ~P z  i^i  Fin )  =  ( ~P A  i^i  Fin ) )
2322rexeqdv 3039 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  ( ~P z  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x
) )
2423abbidv 2565 . . 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 7931 . . 3  |-  fi  =  ( z  e.  _V  |->  { y  |  E. x  e.  ( ~P z  i^i  Fin ) y  =  |^| x }
)
2624, 25fvmptg 5962 . 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 665 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 370    = wceq 1437    e. wcel 1870   {cab 2414    =/= wne 2625   E.wrex 2783   _Vcvv 3087    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   U.cuni 4222   |^|cint 4258   ` cfv 5601   Fincfn 7577   ficfi 7930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-fi 7931
This theorem is referenced by:  elfi  7933  fi0  7940
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