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Theorem iinpw 3989
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 3872 . . . 4  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  C_  x )
2 vex 2752 . . . . . 6  |-  y  e. 
_V
32elpw 3593 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
43ralbii 2493 . . . 4  |-  ( A. x  e.  A  y  e.  ~P x  <->  A. x  e.  A  y  C_  x )
51, 4bitr4i 187 . . 3  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  e.  ~P x )
62elpw 3593 . . 3  |-  ( y  e.  ~P |^| A  <->  y 
C_  |^| A )
7 eliin 3903 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x ) )
82, 7ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x
)
95, 6, 83bitr4i 212 . 2  |-  ( y  e.  ~P |^| A  <->  y  e.  |^|_ x  e.  A  ~P x )
109eqriv 2184 1  |-  ~P |^| A  =  |^|_ x  e.  A  ~P x
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1363    e. wcel 2158   A.wral 2465   _Vcvv 2749    C_ wss 3141   ~Pcpw 3587   |^|cint 3856   |^|_ciin 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589  df-int 3857  df-iin 3901
This theorem is referenced by: (None)
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