Theorem bdcab 16444

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 16433	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 16442	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2217  BOUNDED wbd 16407  BOUNDED wbdc 16435

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 1497  ax-bd0 16408  ax-bdsb 16417

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-bdc 16436

This theorem is referenced by:  bds  16446  bdcrab  16447  bdccsb  16455  bdcdif  16456  bdcun  16457  bdcin  16458  bdcpw  16464  bdcsn  16465  bdcuni  16471  bdcint  16472  bdciun  16473  bdciin  16474  bdcriota  16478  bj-bdfindis  16542