Theorem bdciun 13770

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 ∪ 𝑥 ∈ 𝑦 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.

Step	Hyp	Ref	Expression
1		bdciun.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 13738	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	ax-bdex 13711	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴
4	3	bdcab 13741	. 2 ⊢ BOUNDED {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
5		df-iun 3868	. 2 ⊢ ∪ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
6	4, 5	bdceqir 13736	1 ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴

Syntax hints:   ∈ wcel 2136  {cab 2151  ∃wrex 2445  ∪ ciun 3866  BOUNDED wbdc 13732

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13705  ax-bdex 13711  ax-bdsb 13714

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-iun 3868  df-bdc 13733

This theorem is referenced by: (None)