Theorem viin 4579

Description: Indexed intersection with a universal index class. When A doesn't depend on x, this evaluates to A by 19.3 2069 and abid2 2745. When A = x, this evaluates to (/) by intiin 4574 and intv 4841. (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 4523	. 2 |- |^|_ x e. _V A = { y | A. x e. _V y e. A }
2		ralv 3219	. . 3 |- ( A. x e. _V y e. A <-> A. x y e. A )
3	2	abbii 2739	. 2 |- { y | A. x e. _V y e. A } = { y | A. x y e. A }
4	1, 3	eqtri 2644	1 |- |^|_ x e. _V A = { y | A. x y e. A }

Syntax hints:   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-v 3202  df-iin 4523

This theorem is referenced by: (None)