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Theorem bdciin 15084
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 15051 . . . 4 BOUNDED 𝑧𝐴
32ax-bdal 15023 . . 3 BOUNDED𝑥𝑦 𝑧𝐴
43bdcab 15054 . 2 BOUNDED {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
5 df-iin 3904 . 2 𝑥𝑦 𝐴 = {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
64, 5bdceqir 15049 1 BOUNDED 𝑥𝑦 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2160  {cab 2175  wral 2468   ciin 3902  BOUNDED wbdc 15045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 15018  ax-bdal 15023  ax-bdsb 15027
This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-iin 3904  df-bdc 15046
This theorem is referenced by: (None)
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