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Theorem intiin 3957
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem intiin
StepHypRef Expression
1 dfint2 3865 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 3909 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2307 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff set class
Syntax hints:    = wceq 1624    e. wcel 1685   {cab 2270   A.wral 2544   |^|cint 3863   |^|_ciin 3907
This theorem is referenced by:  relint  4808  intpreima  5617  ixpint  6838  firest  13331  efger  15021  rintopn  16649  intcld  16771  inttop3  24915
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-ral 2549  df-int 3864  df-iin 3909
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