Theorem bdciun 9998

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 9966	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	ax-bdex 9939	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴
4	3	bdcab 9969	. 2 ⊢ BOUNDED {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
5		df-iun 3659	. 2 ⊢ ∪ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
6	4, 5	bdceqir 9964	1 ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴

Syntax hints:   ∈ wcel 1393  {cab 2026  ∃wrex 2307  ∪ ciun 3657  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdex 9939  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-iun 3659  df-bdc 9961

This theorem is referenced by: (None)