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Theorem fival 7166
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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpr 447 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  |^| x )
3 inss1 3389 . . . . . . . . . 10  |-  ( ~P A  i^i  Fin )  C_ 
~P A
43sseli 3176 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
5 elpwi 3633 . . . . . . . . 9  |-  ( x  e.  ~P A  ->  x  C_  A )
64, 5syl 15 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
7 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
8 eleq1 2343 . . . . . . . . . 10  |-  ( y  =  |^| x  -> 
( y  e.  _V  <->  |^| x  e.  _V )
)
97, 8mpbii 202 . . . . . . . . 9  |-  ( y  =  |^| x  ->  |^| x  e.  _V )
10 intex 4167 . . . . . . . . 9  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
119, 10sylibr 203 . . . . . . . 8  |-  ( y  =  |^| x  ->  x  =/=  (/) )
12 intssuni2 3887 . . . . . . . 8  |-  ( ( x  C_  A  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. A )
136, 11, 12syl2an 463 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  |^| x  C_ 
U. A )
142, 13eqsstrd 3212 . . . . . 6  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  C_ 
U. A )
157elpw 3631 . . . . . 6  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
1614, 15sylibr 203 . . . . 5  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  e.  ~P U. A )
1716rexlimiva 2662 . . . 4  |-  ( E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x  ->  y  e.  ~P U. A )
1817abssi 3248 . . 3  |-  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  C_  ~P U. A
19 uniexg 4517 . . . 4  |-  ( A  e.  V  ->  U. A  e.  _V )
20 pwexg 4194 . . . 4  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
2119, 20syl 15 . . 3  |-  ( A  e.  V  ->  ~P U. A  e.  _V )
22 ssexg 4160 . . 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 )
2318, 21, 22sylancr 644 . 2  |-  ( A  e.  V  ->  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  e.  _V )
24 pweq 3628 . . . . . 6  |-  ( z  =  A  ->  ~P z  =  ~P A
)
2524ineq1d 3369 . . . . 5  |-  ( z  =  A  ->  ( ~P z  i^i  Fin )  =  ( ~P A  i^i  Fin ) )
2625rexeqdv 2743 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  ( ~P z  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x
) )
2726abbidv 2397 . . 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 }
)
28 df-fi 7165 . . 3  |-  fi  =  ( z  e.  _V  |->  { y  |  E. x  e.  ( ~P z  i^i  Fin ) y  =  |^| x }
)
2927, 28fvmptg 5600 . 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 } )
301, 23, 29syl2anc 642 1  |-  ( A  e.  V  ->  ( fi `  A )  =  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   ` cfv 5255   Fincfn 6863   ficfi 7164
This theorem is referenced by:  elfi  7167  fi0  7173
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  ax-un 4512
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-int 3863  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-fi 7165
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