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Theorem iinpw 3950
Description: The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
Assertion
Ref Expression
iinpw  |-  ~P |^| A  =  |^|_ x  e.  A  ~P x
Distinct variable group:    x, A

Proof of Theorem iinpw
StepHypRef Expression
1 ssint 3838 . . . 4  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  C_  x )
2 vex 2760 . . . . . 6  |-  y  e. 
_V
32elpw 3591 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
43ralbii 2540 . . . 4  |-  ( A. x  e.  A  y  e.  ~P x  <->  A. x  e.  A  y  C_  x )
51, 4bitr4i 245 . . 3  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  e.  ~P x )
62elpw 3591 . . 3  |-  ( y  e.  ~P |^| A  <->  y 
C_  |^| A )
7 eliin 3870 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x ) )
82, 7ax-mp 10 . . 3  |-  ( y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x
)
95, 6, 83bitr4i 270 . 2  |-  ( y  e.  ~P |^| A  <->  y  e.  |^|_ x  e.  A  ~P x )
109eqriv 2253 1  |-  ~P |^| A  =  |^|_ x  e.  A  ~P x
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2757    C_ wss 3113   ~Pcpw 3585   |^|cint 3822   |^|_ciin 3866
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2521  df-v 2759  df-in 3120  df-ss 3127  df-pw 3587  df-int 3823  df-iin 3868
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