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Theorem intiin 3837
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 3743 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 3786 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2141 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff set class
Syntax hints:    = wceq 1316   {cab 2103   A.wral 2393   |^|cint 3741   |^|_ciin 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-ral 2398  df-int 3742  df-iin 3786
This theorem is referenced by:  relint  4633  ixpintm  6587
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