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Theorem dfiin2 3938
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 3936 . 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 10 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2612 1  |-  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   |^|cint 3862   |^|_ciin 3906
This theorem is referenced by:  fniinfv  5581  scott0  7556  cfval2  7886  cflim3  7888  cflim2  7889  cfss  7891  hauscmplem  17133  ptbasfi  17276  dihglblem5  31488  dihglb2  31532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-int 3863  df-iin 3908
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