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Theorem bdciin 11653
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 𝑥𝑦 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 11620 . . . 4 BOUNDED 𝑧𝐴
32ax-bdal 11592 . . 3 BOUNDED𝑥𝑦 𝑧𝐴
43bdcab 11623 . 2 BOUNDED {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
5 df-iin 3731 . 2 𝑥𝑦 𝐴 = {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
64, 5bdceqir 11618 1 BOUNDED 𝑥𝑦 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1438  {cab 2074  wral 2359   ciin 3729  BOUNDED wbdc 11614
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11587  ax-bdal 11592  ax-bdsb 11596
This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-iin 3731  df-bdc 11615
This theorem is referenced by: (None)
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