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Theorem fival 7419
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 2966 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpr 449 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  =  |^| x )
3 inss1 3563 . . . . . . . . . 10  |-  ( ~P A  i^i  Fin )  C_ 
~P A
43sseli 3346 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
54elpwid 3810 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
6 vex 2961 . . . . . . . . . 10  |-  y  e. 
_V
7 eleq1 2498 . . . . . . . . . 10  |-  ( y  =  |^| x  -> 
( y  e.  _V  <->  |^| x  e.  _V )
)
86, 7mpbii 204 . . . . . . . . 9  |-  ( y  =  |^| x  ->  |^| x  e.  _V )
9 intex 4358 . . . . . . . . 9  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
108, 9sylibr 205 . . . . . . . 8  |-  ( y  =  |^| x  ->  x  =/=  (/) )
11 intssuni2 4077 . . . . . . . 8  |-  ( ( x  C_  A  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. A )
125, 10, 11syl2an 465 . . . . . . 7  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  |^| x  C_ 
U. A )
132, 12eqsstrd 3384 . . . . . 6  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  C_ 
U. A )
146elpw 3807 . . . . . 6  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
1513, 14sylibr 205 . . . . 5  |-  ( ( x  e.  ( ~P A  i^i  Fin )  /\  y  =  |^| x )  ->  y  e.  ~P U. A )
1615rexlimiva 2827 . . . 4  |-  ( E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x  ->  y  e.  ~P U. A )
1716abssi 3420 . . 3  |-  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  C_  ~P U. A
18 uniexg 4708 . . . 4  |-  ( A  e.  V  ->  U. A  e.  _V )
19 pwexg 4385 . . . 4  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
2018, 19syl 16 . . 3  |-  ( A  e.  V  ->  ~P U. A  e.  _V )
21 ssexg 4351 . . 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 )
2217, 20, 21sylancr 646 . 2  |-  ( A  e.  V  ->  { y  |  E. x  e.  ( ~P A  i^i  Fin ) y  =  |^| x }  e.  _V )
23 pweq 3804 . . . . . 6  |-  ( z  =  A  ->  ~P z  =  ~P A
)
2423ineq1d 3543 . . . . 5  |-  ( z  =  A  ->  ( ~P z  i^i  Fin )  =  ( ~P A  i^i  Fin ) )
2524rexeqdv 2913 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  ( ~P z  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P A  i^i  Fin )
y  =  |^| x
) )
2625abbidv 2552 . . 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 }
)
27 df-fi 7418 . . 3  |-  fi  =  ( z  e.  _V  |->  { y  |  E. x  e.  ( ~P z  i^i  Fin ) y  =  |^| x }
)
2826, 27fvmptg 5806 . 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 } )
291, 22, 28syl2anc 644 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 360    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   U.cuni 4017   |^|cint 4052   ` cfv 5456   Fincfn 7111   ficfi 7417
This theorem is referenced by:  elfi  7420  fi0  7427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-fi 7418
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