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Theorem intiin 4372
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 4277 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 4321 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2492 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   {cab 2445   A.wral 2807   |^|cint 4275   |^|_ciin 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-ral 2812  df-int 4276  df-iin 4321
This theorem is referenced by:  relint  5117  intpreima  6003  ixpint  7486  firest  14677  efger  16525  rintopn  19178  intcld  19300  iundifdifd  27088  iundifdif  27089
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