Theorem bdciin 16905

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 16872	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	ax-bdal 16844	. . 3 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴
4	3	bdcab 16875	. 2 ⊢ BOUNDED {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
5		df-iin 4015	. 2 ⊢ ∩ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴}
6	4, 5	bdceqir 16870	1 ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴

This proof depends on syntax axioms:   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∩ ciin 4013  BOUNDED wbdc 16866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16839  ax-bdal 16844  ax-bdsb 16848

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-iin 4015  df-bdc 16867

This theorem is used by: (None)