Theorem bdcint 16696

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 16640	. . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧
2	1	ax-bdal 16637	. . . 4 ⊢ BOUNDED ∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧
3		df-ral 2527	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧))
4	2, 3	bd0 16643	. . 3 ⊢ BOUNDED ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)
5	4	bdcab 16668	. 2 ⊢ BOUNDED {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
6		df-int 3952	. 2 ⊢ ∩ 𝑥 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)}
7	5, 6	bdceqir 16663	1 ⊢ BOUNDED ∩ 𝑥

Syntax hints:   → wi 4  ∀wal 1396  {cab 2220  ∀wral 2522  ∩ cint 3951  BOUNDED wbdc 16659

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216  ax-bd0 16632  ax-bdal 16637  ax-bdel 16640  ax-bdsb 16641

This theorem depends on definitions:  df-bi 117  df-clab 2221  df-cleq 2227  df-clel 2230  df-ral 2527  df-int 3952  df-bdc 16660

This theorem is referenced by: (None)