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Theorem bdciun 10827
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 10795 . . . 4 BOUNDED 𝑧𝐴
32ax-bdex 10768 . . 3 BOUNDED𝑥𝑦 𝑧𝐴
43bdcab 10798 . 2 BOUNDED {𝑧 ∣ ∃𝑥𝑦 𝑧𝐴}
5 df-iun 3688 . 2 𝑥𝑦 𝐴 = {𝑧 ∣ ∃𝑥𝑦 𝑧𝐴}
64, 5bdceqir 10793 1 BOUNDED 𝑥𝑦 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1434  {cab 2068  wrex 2350   ciun 3686  BOUNDED wbdc 10789
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064  ax-bd0 10762  ax-bdex 10768  ax-bdsb 10771
This theorem depends on definitions:  df-bi 115  df-clab 2069  df-cleq 2075  df-clel 2078  df-iun 3688  df-bdc 10790
This theorem is referenced by: (None)
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