Theorem bdcint 15307

Description: The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)

Assertion

Ref	Expression
bdcint	⊢ BOUNDED ∩ 𝑥

Proof of Theorem bdcint

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-bdel 15251	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdal 15248	. . . 4 ⊢ BOUNDED ∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3		df-ral 2477	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))
4	2, 3	bd0 15254	. . 3 ⊢ BOUNDED ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)
5	4	bdcab 15279	. 2 ⊢ BOUNDED {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
6		df-int 3871	. 2 ⊢ ∩ 𝑥 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
7	5, 6	bdceqir 15274	1 ⊢ BOUNDED ∩ 𝑥

Syntax hints:   → wi 4  ∀wal 1362  {cab 2179  ∀wral 2472  ∩ cint 3870  BOUNDED wbdc 15270

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175  ax-bd0 15243  ax-bdal 15248  ax-bdel 15251  ax-bdsb 15252

This theorem depends on definitions:  df-bi 117  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-int 3871  df-bdc 15271

This theorem is referenced by: (None)