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Theorem dfiin2 3912
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 3910 . 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 12 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2587 1  |-  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   {cab 2244   E.wrex 2519   _Vcvv 2763   |^|cint 3836   |^|_ciin 3880
This theorem is referenced by:  fniinfv  5515  scott0  7524  cfval2  7854  cflim3  7856  cflim2  7857  cfss  7859  hauscmplem  17095  ptbasfi  17238  dihglblem5  30655  dihglb2  30699
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-v 2765  df-int 3837  df-iin 3882
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