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Theorem bdciin 12900
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 12867 . . . 4 BOUNDED 𝑧𝐴
32ax-bdal 12839 . . 3 BOUNDED𝑥𝑦 𝑧𝐴
43bdcab 12870 . 2 BOUNDED {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
5 df-iin 3784 . 2 𝑥𝑦 𝐴 = {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
64, 5bdceqir 12865 1 BOUNDED 𝑥𝑦 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1463  {cab 2101  wral 2391   ciin 3782  BOUNDED wbdc 12861
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097  ax-bd0 12834  ax-bdal 12839  ax-bdsb 12843
This theorem depends on definitions:  df-bi 116  df-clab 2102  df-cleq 2108  df-clel 2111  df-iin 3784  df-bdc 12862
This theorem is referenced by: (None)
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