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Theorem bdciin 9999
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 9966 . . . 4  |- BOUNDED  z  e.  A
32ax-bdal 9938 . . 3  |- BOUNDED  A. x  e.  y  z  e.  A
43bdcab 9969 . 2  |- BOUNDED  { z  |  A. x  e.  y  z  e.  A }
5 df-iin 3660 . 2  |-  |^|_ x  e.  y  A  =  { z  |  A. x  e.  y  z  e.  A }
64, 5bdceqir 9964 1  |- BOUNDED 
|^|_ x  e.  y  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1393   {cab 2026   A.wral 2306   |^|_ciin 3658  BOUNDED wbdc 9960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdal 9938  ax-bdsb 9942
This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-iin 3660  df-bdc 9961
This theorem is referenced by: (None)
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