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Theorem intiin 3916
Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
intiin  |-  |^| A  =  |^|_ x  e.  A  x
Distinct variable group:    x, A

Proof of Theorem intiin
StepHypRef Expression
1 dfint2 3824 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 3868 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2279 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   {cab 2242   A.wral 2516   |^|cint 3822   |^|_ciin 3866
This theorem is referenced by:  relint  4783  intpreima  5576  ixpint  6797  firest  13285  efger  14975  rintopn  16603  intcld  16725  inttop3  24869
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 2237
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-ral 2521  df-int 3823  df-iin 3868
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