Theorem bdcrab 15750

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 15744	. . . 4 |- BOUNDED x e. A
3		bdcrab.2	. . . 4 |- BOUNDED ph
4	2, 3	ax-bdan 15713	. . 3 |- BOUNDED ( x e. A /\ ph )
5	4	bdcab 15747	. 2 |- BOUNDED { x | ( x e. A /\ ph ) }
6		df-rab 2492	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	5, 6	bdceqir 15742	1 |- BOUNDED { x e. A | ph }

Syntax hints:   /\ wa 104   e. wcel 2175   {cab 2190   {crab 2487  BOUNDED wbd 15710  BOUNDED wbdc 15738

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-bd0 15711  ax-bdan 15713  ax-bdsb 15720

This theorem depends on definitions:  df-bi 117  df-clab 2191  df-cleq 2197  df-clel 2200  df-rab 2492  df-bdc 15739

This theorem is referenced by:  bdrabexg  15804