Theorem viin 4030

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 3973	. 2 |- |^|_ x e. _V A = { y | A. x e. _V y e. A }
2		ralv 2820	. . 3 |- ( A. x e. _V y e. A <-> A. x y e. A )
3	2	abbii 2347	. 2 |- { y | A. x e. _V y e. A } = { y | A. x y e. A }
4	1, 3	eqtri 2252	1 |- |^|_ x e. _V A = { y | A. x y e. A }

Syntax hints:   A.wal 1395   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   _Vcvv 2802   |^|_ciin 3971

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-v 2804  df-iin 3973

This theorem is referenced by: (None)