Theorem viin 3948

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 3891	. 2 |- |^|_ x e. _V A = { y | A. x e. _V y e. A }
2		ralv 2756	. . 3 |- ( A. x e. _V y e. A <-> A. x y e. A )
3	2	abbii 2293	. 2 |- { y | A. x e. _V y e. A } = { y | A. x y e. A }
4	1, 3	eqtri 2198	1 |- |^|_ x e. _V A = { y | A. x y e. A }

Syntax hints:   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   _Vcvv 2739   |^|_ciin 3889

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-v 2741  df-iin 3891

This theorem is referenced by: (None)