Theorem bdcab 16170

Description: A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdcab.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdcab	⊢ BOUNDED {𝑥 ∣ 𝜑}

Proof of Theorem bdcab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcab.1	. . 3 ⊢ BOUNDED 𝜑
2	1	bdab 16159	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 16168	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2215  BOUNDED wbd 16133  BOUNDED wbdc 16161

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-gen 1495  ax-bd0 16134  ax-bdsb 16143

This theorem depends on definitions:  df-bi 117  df-clab 2216  df-bdc 16162

This theorem is referenced by:  bds  16172  bdcrab  16173  bdccsb  16181  bdcdif  16182  bdcun  16183  bdcin  16184  bdcpw  16190  bdcsn  16191  bdcuni  16197  bdcint  16198  bdciun  16199  bdciin  16200  bdcriota  16204  bj-bdfindis  16268