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Theorem viin 4384
Description: Indexed intersection with a universal index class. When  A doesn't depend on  x, this evaluates to  A by 19.3 1836 and abid2 2607. When  A  =  x, this evaluates to  (/) by intiin 4379 and intv 4623. (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 4328 . 2  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  e.  _V  y  e.  A }
2 ralv 3127 . . 3  |-  ( A. x  e.  _V  y  e.  A  <->  A. x  y  e.  A )
32abbii 2601 . 2  |-  { y  |  A. x  e. 
_V  y  e.  A }  =  { y  |  A. x  y  e.  A }
41, 3eqtri 2496 1  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  y  e.  A }
Colors of variables: wff setvar class
Syntax hints:   A.wal 1377    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113   |^|_ciin 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2819  df-v 3115  df-iin 4328
This theorem is referenced by: (None)
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