Theorem bdcrab 13734

Description: A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdcrab.1	|- BOUNDED A
bdcrab.2	|- BOUNDED ph

Assertion

Ref	Expression
bdcrab	|- BOUNDED { x e. A | ph }

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem bdcrab

Step	Hyp	Ref	Expression
1		bdcrab.1	. . . . 5 |- BOUNDED A
2	1	bdeli 13728	. . . 4 |- BOUNDED x e. A
3		bdcrab.2	. . . 4 |- BOUNDED ph
4	2, 3	ax-bdan 13697	. . 3 |- BOUNDED ( x e. A /\ ph )
5	4	bdcab 13731	. 2 |- BOUNDED { x | ( x e. A /\ ph ) }
6		df-rab 2453	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	5, 6	bdceqir 13726	1 |- BOUNDED { x e. A | ph }

Syntax hints:   /\ wa 103   e. wcel 2136   {cab 2151   {crab 2448  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-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13695  ax-bdan 13697  ax-bdsb 13704

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-rab 2453  df-bdc 13723

This theorem is referenced by:  bdrabexg  13788