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Theorem iinpw 4139
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 4026 . . . 4  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  C_  x )
2 vex 2919 . . . . . 6  |-  y  e. 
_V
32elpw 3765 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
43ralbii 2690 . . . 4  |-  ( A. x  e.  A  y  e.  ~P x  <->  A. x  e.  A  y  C_  x )
51, 4bitr4i 244 . . 3  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  e.  ~P x )
62elpw 3765 . . 3  |-  ( y  e.  ~P |^| A  <->  y 
C_  |^| A )
7 eliin 4058 . . . 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 269 . 2  |-  ( y  e.  ~P |^| A  <->  y  e.  |^|_ x  e.  A  ~P x )
109eqriv 2401 1  |-  ~P |^| A  =  |^|_ x  e.  A  ~P x
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   |^|cint 4010   |^|_ciin 4054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-in 3287  df-ss 3294  df-pw 3761  df-int 4011  df-iin 4056
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