Theorem bdcab 16611

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 16600	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 16609	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2218  BOUNDED wbd 16574  BOUNDED wbdc 16602

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 16575  ax-bdsb 16584

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-bdc 16603

This theorem is referenced by:  bds  16613  bdcrab  16614  bdccsb  16622  bdcdif  16623  bdcun  16624  bdcin  16625  bdcpw  16631  bdcsn  16632  bdcuni  16638  bdcint  16639  bdciun  16640  bdciin  16641  bdcriota  16645  bj-bdfindis  16709