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Theorem viin 4330
Description: Indexed intersection with a universal index class. When  A doesn't depend on  x, this evaluates to  A by 19.3 1912 and abid2 2542. When  A  =  x, this evaluates to  (/) by intiin 4325 and intv 4570. (Contributed by NM, 11-Sep-2008.)
Assertion
Ref Expression
viin  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  y  e.  A }
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem viin
StepHypRef Expression
1 df-iin 4274 . 2  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  e.  _V  y  e.  A }
2 ralv 3073 . . 3  |-  ( A. x  e.  _V  y  e.  A  <->  A. x  y  e.  A )
32abbii 2536 . 2  |-  { y  |  A. x  e. 
_V  y  e.  A }  =  { y  |  A. x  y  e.  A }
41, 3eqtri 2431 1  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  y  e.  A }
Colors of variables: wff setvar class
Syntax hints:   A.wal 1403    = wceq 1405    e. wcel 1842   {cab 2387   A.wral 2754   _Vcvv 3059   |^|_ciin 4272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-ral 2759  df-v 3061  df-iin 4274
This theorem is referenced by: (None)
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