Theorem viin 3976

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 3919	. 2 |- |^|_ x e. _V A = { y | A. x e. _V y e. A }
2		ralv 2780	. . 3 |- ( A. x e. _V y e. A <-> A. x y e. A )
3	2	abbii 2312	. 2 |- { y | A. x e. _V y e. A } = { y | A. x y e. A }
4	1, 3	eqtri 2217	1 |- |^|_ x e. _V A = { y | A. x y e. A }

Syntax hints:   A.wal 1362   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   _Vcvv 2763   |^|_ciin 3917

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-v 2765  df-iin 3919

This theorem is referenced by: (None)