Theorem bdciun 16774

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 16742	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	ax-bdex 16715	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴
4	3	bdcab 16745	. 2 ⊢ BOUNDED {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
5		df-iun 3998	. 2 ⊢ ∪ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
6	4, 5	bdceqir 16740	1 ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴

Syntax hints:   ∈ wcel 2205  {cab 2220  ∃wrex 2523  ∪ ciun 3996  BOUNDED wbdc 16736

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216  ax-bd0 16709  ax-bdex 16715  ax-bdsb 16718

This theorem depends on definitions:  df-bi 117  df-clab 2221  df-cleq 2227  df-clel 2230  df-iun 3998  df-bdc 16737

This theorem is referenced by: (None)