Theorem bdciin 10670

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 10637	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	ax-bdal 10609	. . 3 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴
4	3	bdcab 10640	. 2 ⊢ BOUNDED {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
5		df-iin 3681	. 2 ⊢ ∩ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
6	4, 5	bdceqir 10635	1 ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴

Syntax hints:   ∈ wcel 1433  {cab 2067  ∀wral 2348  ∩ ciin 3679  BOUNDED wbdc 10631

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604  ax-bdal 10609  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-iin 3681  df-bdc 10632

This theorem is referenced by: (None)