Theorem viin 3987

Description: Indexed intersection with a universal index class. (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

Step	Hyp	Ref	Expression
1		df-iin 3930	. 2 |- |^|_ x e. _V A = { y | A. x e. _V y e. A }
2		ralv 2789	. . 3 |- ( A. x e. _V y e. A <-> A. x y e. A )
3	2	abbii 2321	. 2 |- { y | A. x e. _V y e. A } = { y | A. x y e. A }
4	1, 3	eqtri 2226	1 |- |^|_ x e. _V A = { y | A. x y e. A }

Syntax hints:   A.wal 1371   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   _Vcvv 2772   |^|_ciin 3928

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-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-ral 2489  df-v 2774  df-iin 3930

This theorem is referenced by: (None)