Theorem bdcrab 16792

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 16786	. . . 4 |- BOUNDED x e. A
3		bdcrab.2	. . . 4 |- BOUNDED ph
4	2, 3	ax-bdan 16755	. . 3 |- BOUNDED ( x e. A /\ ph )
5	4	bdcab 16789	. 2 |- BOUNDED { x | ( x e. A /\ ph ) }
6		df-rab 2537	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	5, 6	bdceqir 16784	1 |- BOUNDED { x e. A | ph }

Syntax hints:   /\ wa 104   e. wcel 2209   {cab 2224   {crab 2532  BOUNDED wbd 16752  BOUNDED wbdc 16780

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-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16753  ax-bdan 16755  ax-bdsb 16762

This theorem depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-rab 2537  df-bdc 16781

This theorem is referenced by:  bdrabexg  16846