Theorem bdcab 16495

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 16484	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 16493	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2217  BOUNDED wbd 16458  BOUNDED wbdc 16486

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 16459  ax-bdsb 16468

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-bdc 16487

This theorem is referenced by:  bds  16497  bdcrab  16498  bdccsb  16506  bdcdif  16507  bdcun  16508  bdcin  16509  bdcpw  16515  bdcsn  16516  bdcuni  16522  bdcint  16523  bdciun  16524  bdciin  16525  bdcriota  16529  bj-bdfindis  16593