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Theorem dfiin2 3765
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 3763 . 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 9 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2432 1  |-  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619   |^|cint 3688   |^|_ciin 3731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-int 3689  df-iin 3733
This theorem is referenced by: (None)
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