Theorem dfiin2 4205

Description: Alternate definition of indexed intersection when B is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Hypothesis

Ref	Expression
dfiun2.1	|- B e. _V

Assertion

Ref	Expression
dfiin2	|- |^|_ x e. A B = |^| { y | E. x e. A y = B }

Distinct variable groups:   x, y   y, A   y, B

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem dfiin2

Step	Hyp	Ref	Expression
1		dfiin2g 4203	. 2 |- ( A. x e. A B e. _V -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
2		dfiun2.1	. . 3 |- B e. _V
3	2	a1i 11	. 2 |- ( x e. A -> B e. _V )
4	1, 3	mprg 2785	1 |- |^|_ x e. A B = |^| { y | E. x e. A y = B }

Syntax hints:   = wceq 1369   e. wcel 1756   {cab 2429   E.wrex 2716   _Vcvv 2972   |^|cint 4128   |^|_ciin 4172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-v 2974  df-int 4129  df-iin 4174

This theorem is referenced by:  fniinfv  5750  scott0  8093  cfval2  8429  cflim3  8431  cflim2  8432  cfss  8434  hauscmplem  19009  ptbasfi  19154  dihglblem5  34943  dihglb2  34987