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Theorem bdciin 10921
Description: The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdciun.1  |- BOUNDED  A
Assertion
Ref Expression
bdciin  |- BOUNDED 
|^|_ x  e.  y  A
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem bdciin
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bdciun.1 . . . . 5  |- BOUNDED  A
21bdeli 10888 . . . 4  |- BOUNDED  z  e.  A
32ax-bdal 10860 . . 3  |- BOUNDED  A. x  e.  y  z  e.  A
43bdcab 10891 . 2  |- BOUNDED  { z  |  A. x  e.  y  z  e.  A }
5 df-iin 3702 . 2  |-  |^|_ x  e.  y  A  =  { z  |  A. x  e.  y  z  e.  A }
64, 5bdceqir 10886 1  |- BOUNDED 
|^|_ x  e.  y  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   {cab 2069   A.wral 2353   |^|_ciin 3700  BOUNDED wbdc 10882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065  ax-bd0 10855  ax-bdal 10860  ax-bdsb 10864
This theorem depends on definitions:  df-bi 115  df-clab 2070  df-cleq 2076  df-clel 2079  df-iin 3702  df-bdc 10883
This theorem is referenced by: (None)
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