Theorem bdciin 16014

Description: The indexed intersection 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
bdciin	⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem bdciin

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdciun.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 15981	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	ax-bdal 15953	. . 3 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴
4	3	bdcab 15984	. 2 ⊢ BOUNDED {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
5		df-iin 3944	. 2 ⊢ ∩ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
6	4, 5	bdceqir 15979	1 ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴

Syntax hints:   ∈ wcel 2178  {cab 2193  ∀wral 2486  ∩ ciin 3942  BOUNDED wbdc 15975

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-bd0 15948  ax-bdal 15953  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-clab 2194  df-cleq 2200  df-clel 2203  df-iin 3944  df-bdc 15976

This theorem is referenced by: (None)