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Theorem iinpw 4391
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 4271 . . . 4  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  C_  x )
2 selpw 3988 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
32ralbii 2853 . . . 4  |-  ( A. x  e.  A  y  e.  ~P x  <->  A. x  e.  A  y  C_  x )
41, 3bitr4i 255 . . 3  |-  ( y 
C_  |^| A  <->  A. x  e.  A  y  e.  ~P x )
5 selpw 3988 . . 3  |-  ( y  e.  ~P |^| A  <->  y 
C_  |^| A )
6 vex 3083 . . . 4  |-  y  e. 
_V
7 eliin 4305 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x ) )
86, 7ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  A  ~P x  <->  A. x  e.  A  y  e.  ~P x
)
94, 5, 83bitr4i 280 . 2  |-  ( y  e.  ~P |^| A  <->  y  e.  |^|_ x  e.  A  ~P x )
109eqriv 2418 1  |-  ~P |^| A  =  |^|_ x  e.  A  ~P x
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1872   A.wral 2771   _Vcvv 3080    C_ wss 3436   ~Pcpw 3981   |^|cint 4255   |^|_ciin 4300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-v 3082  df-in 3443  df-ss 3450  df-pw 3983  df-int 4256  df-iin 4302
This theorem is referenced by: (None)
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