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Theorem intiin 4350
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 4254 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 4299 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2454 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   {cab 2407   A.wral 2775   |^|cint 4252   |^|_ciin 4297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-ral 2780  df-int 4253  df-iin 4299
This theorem is referenced by:  relint  4973  intpreima  6023  ixpint  7554  firest  15319  efger  17356  rintopn  19926  intcld  20042  iundifdifd  28167  iundifdif  28168
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