Theorem bdcint 16886

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 16830	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdal 16827	. . . 4 ⊢ BOUNDED ∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3		df-ral 2533	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))
4	2, 3	bd0 16833	. . 3 ⊢ BOUNDED ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)
5	4	bdcab 16858	. 2 ⊢ BOUNDED {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
6		df-int 3969	. 2 ⊢ ∩ 𝑥 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
7	5, 6	bdceqir 16853	1 ⊢ BOUNDED ∩ 𝑥

Syntax hints:   → wi 4  ∀wal 1400  {cab 2224  ∀wral 2528  ∩ cint 3968  BOUNDED wbdc 16849

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16822  ax-bdal 16827  ax-bdel 16830  ax-bdsb 16831

This theorem depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-int 3969  df-bdc 16850

This theorem is referenced by: (None)