Theorem bdcrab 11188

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 11182	. . . 4 |- BOUNDED x e. A
3		bdcrab.2	. . . 4 |- BOUNDED ph
4	2, 3	ax-bdan 11151	. . 3 |- BOUNDED ( x e. A /\ ph )
5	4	bdcab 11185	. 2 |- BOUNDED { x | ( x e. A /\ ph ) }
6		df-rab 2364	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	5, 6	bdceqir 11180	1 |- BOUNDED { x e. A | ph }

Syntax hints:   /\ wa 102   e. wcel 1436   {cab 2071   {crab 2359  BOUNDED wbd 11148  BOUNDED wbdc 11176

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067  ax-bd0 11149  ax-bdan 11151  ax-bdsb 11158

This theorem depends on definitions:  df-bi 115  df-clab 2072  df-cleq 2078  df-clel 2081  df-rab 2364  df-bdc 11177

This theorem is referenced by:  bdrabexg  11242