Theorem viin 4026

Description: Indexed intersection with a universal index class. When A doesn't depend on x, this evaluates to A by 19.3 1785 and abid2 2471. When A = x, this evaluates to (/) by intiin 4021 and intv in set.mm. (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 2873	. . 3 |- (A.x e. _V y e. A <-> A.x y e. A)
3	2	abbii 2466	. 2 |- {y | A.x e. _V y e. A} = {y | A.x y e. A}
4	1, 3	eqtri 2373	1 |- |^|_x e. _V A = {y | A.x y e. A}

Syntax hints:  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  _Vcvv 2860  |^|_ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-v 2862  df-iin 3973

This theorem is referenced by: (None)