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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
StepHypRef 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
32a1i 11 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2795 1  |-  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff setvar class
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
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