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Theorem bdciun 14252
Description: The indexed union 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
bdciun BOUNDED 𝑥𝑦 𝐴
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem bdciun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bdciun.1 . . . . 5 BOUNDED 𝐴
21bdeli 14220 . . . 4 BOUNDED 𝑧𝐴
32ax-bdex 14193 . . 3 BOUNDED𝑥𝑦 𝑧𝐴
43bdcab 14223 . 2 BOUNDED {𝑧 ∣ ∃𝑥𝑦 𝑧𝐴}
5 df-iun 3886 . 2 𝑥𝑦 𝐴 = {𝑧 ∣ ∃𝑥𝑦 𝑧𝐴}
64, 5bdceqir 14218 1 BOUNDED 𝑥𝑦 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2148  {cab 2163  wrex 2456   ciun 3884  BOUNDED wbdc 14214
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14187  ax-bdex 14193  ax-bdsb 14196
This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-iun 3886  df-bdc 14215
This theorem is referenced by: (None)
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