Theorem bdciin 15525

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 15492	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	ax-bdal 15464	. . 3 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴
4	3	bdcab 15495	. 2 ⊢ BOUNDED {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
5		df-iin 3919	. 2 ⊢ ∩ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
6	4, 5	bdceqir 15490	1 ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴

Syntax hints:   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∩ ciin 3917  BOUNDED wbdc 15486

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15459  ax-bdal 15464  ax-bdsb 15468

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-iin 3919  df-bdc 15487

This theorem is referenced by: (None)