Theorem bdcint 16012

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 15956	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdal 15953	. . . 4 ⊢ BOUNDED ∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3		df-ral 2491	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))
4	2, 3	bd0 15959	. . 3 ⊢ BOUNDED ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)
5	4	bdcab 15984	. 2 ⊢ BOUNDED {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
6		df-int 3900	. 2 ⊢ ∩ 𝑥 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
7	5, 6	bdceqir 15979	1 ⊢ BOUNDED ∩ 𝑥

Syntax hints:   → wi 4  ∀wal 1371  {cab 2193  ∀wral 2486  ∩ cint 3899  BOUNDED wbdc 15975

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-bd0 15948  ax-bdal 15953  ax-bdel 15956  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-int 3900  df-bdc 15976

This theorem is referenced by: (None)