Theorem bdcrab 13221

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 13215	. . . 4 |- BOUNDED x e. A
3		bdcrab.2	. . . 4 |- BOUNDED ph
4	2, 3	ax-bdan 13184	. . 3 |- BOUNDED ( x e. A /\ ph )
5	4	bdcab 13218	. 2 |- BOUNDED { x | ( x e. A /\ ph ) }
6		df-rab 2426	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	5, 6	bdceqir 13213	1 |- BOUNDED { x e. A | ph }

Syntax hints:   /\ wa 103   e. wcel 1481   {cab 2126   {crab 2421  BOUNDED wbd 13181  BOUNDED wbdc 13209

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122  ax-bd0 13182  ax-bdan 13184  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-clab 2127  df-cleq 2133  df-clel 2136  df-rab 2426  df-bdc 13210

This theorem is referenced by:  bdrabexg  13275