Theorem bdcab 16619

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 16608	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 16617	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2218  BOUNDED wbd 16582  BOUNDED wbdc 16610

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 1498  ax-bd0 16583  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-bdc 16611

This theorem is referenced by:  bds  16621  bdcrab  16622  bdccsb  16630  bdcdif  16631  bdcun  16632  bdcin  16633  bdcpw  16639  bdcsn  16640  bdcuni  16646  bdcint  16647  bdciun  16648  bdciin  16649  bdcriota  16653  bj-bdfindis  16717