Theorem bdcab 13731

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 13720	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 13729	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2151  BOUNDED wbd 13694  BOUNDED wbdc 13722

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 1437  ax-bd0 13695  ax-bdsb 13704

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-bdc 13723

This theorem is referenced by:  bds  13733  bdcrab  13734  bdccsb  13742  bdcdif  13743  bdcun  13744  bdcin  13745  bdcpw  13751  bdcsn  13752  bdcuni  13758  bdcint  13759  bdciun  13760  bdciin  13761  bdcriota  13765  bj-bdfindis  13829