Theorem dfiin2 4002

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 4000	. 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 10	. 2 |- (x e. A -> B e. _V)
4	1, 3	mprg 2683	1 |- |^|_x e. A B = |^|{y | E.x e. A y = B}

Syntax hints:   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  _Vcvv 2859  |^|cint 3926  |^|_ciin 3970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-or 359  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-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-int 3927  df-iin 3972

This theorem is referenced by:  fniinfv  5372