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Theorem intiin 4346
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfint2 4250 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 4295 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2487 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455   {cab 2448   A.wral 2749   |^|cint 4248   |^|_ciin 4293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-ral 2754  df-int 4249  df-iin 4295
This theorem is referenced by:  relint  4976  intpreima  6034  ixpint  7575  firest  15380  efger  17417  rintopn  19988  intcld  20104  iundifdifd  28226  iundifdif  28227
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