Theorem bdcab 13849

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 13838	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 13847	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2156  BOUNDED wbd 13812  BOUNDED wbdc 13840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-bd0 13813  ax-bdsb 13822

This theorem depends on definitions:  df-bi 116  df-clab 2157  df-bdc 13841

This theorem is referenced by:  bds  13851  bdcrab  13852  bdccsb  13860  bdcdif  13861  bdcun  13862  bdcin  13863  bdcpw  13869  bdcsn  13870  bdcuni  13876  bdcint  13877  bdciun  13878  bdciin  13879  bdcriota  13883  bj-bdfindis  13947