Theorem dfiin2 4555

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 4553	. 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 2926	1 |- |^|_ x e. A B = |^| { y | E. x e. A y = B }

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   |^|cint 4475   |^|_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-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-int 4476  df-iin 4523

This theorem is referenced by:  fniinfv  6257  scott0  8749  cfval2  9082  cflim3  9084  cflim2  9085  cfss  9087  hauscmplem  21209  ptbasfi  21384  dihglblem5  36587  dihglb2  36631  intima0  37939