Theorem dfiin2 4337

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 4335	. 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 2795	1 |- |^|_ x e. A B = |^| { y | E. x e. A y = B }

Syntax hints:   = wceq 1437   e. wcel 1870   {cab 2414   E.wrex 2783   _Vcvv 3087   |^|cint 4258   |^|_ciin 4303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-v 3089  df-int 4259  df-iin 4305

This theorem is referenced by:  fniinfv  5940  scott0  8356  cfval2  8688  cflim3  8690  cflim2  8691  cfss  8693  hauscmplem  20352  ptbasfi  20527  dihglblem5  34575  dihglb2  34619  intima0  35878