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Theorem iinpw 3990
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssint 3878 . . . 4  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  C_  x )
2 vex 2791 . . . . . 6  |-  y  e. 
_V
32elpw 3631 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
43ralbii 2567 . . . 4  |-  ( A. x  e.  A  y  e.  ~P x  <->  A. x  e.  A  y  C_  x )
51, 4bitr4i 243 . . 3  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  e.  ~P x )
62elpw 3631 . . 3  |-  ( y  e.  ~P |^| A  <->  y 
C_  |^| A )
7 eliin 3910 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x ) )
82, 7ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x
)
95, 6, 83bitr4i 268 . 2  |-  ( y  e.  ~P |^| A  <->  y  e.  |^|_ x  e.  A  ~P x )
109eqriv 2280 1  |-  ~P |^| A  =  |^|_ x  e.  A  ~P x
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   |^|cint 3862   |^|_ciin 3906
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-int 3863  df-iin 3908
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