Theorem viin 3867

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 3811	. 2 |- |^|_ x e. _V A = { y | A. x e. _V y e. A }
2		ralv 2698	. . 3 |- ( A. x e. _V y e. A <-> A. x y e. A )
3	2	abbii 2253	. 2 |- { y | A. x e. _V y e. A } = { y | A. x y e. A }
4	1, 3	eqtri 2158	1 |- |^|_ x e. _V A = { y | A. x y e. A }

Syntax hints:   A.wal 1329   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   _Vcvv 2681   |^|_ciin 3809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-v 2683  df-iin 3811

This theorem is referenced by: (None)