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Theorem dfiin2 3683
 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 V
Assertion
Ref Expression
dfiin2 x A B = {yx 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 3681 . 2 (x A B V → x A B = {yx A y = B})
2 dfiun2.1 . . 3 B V
32a1i 9 . 2 (x AB V)
41, 3mprg 2372 1 x A B = {yx A y = B}
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551  ∩ cint 3606  ∩ ciin 3649 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-int 3607  df-iin 3651 This theorem is referenced by: (None)
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