Theorem bdcrab 16447

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 16441	. . . 4 |- BOUNDED x e. A
3		bdcrab.2	. . . 4 |- BOUNDED ph
4	2, 3	ax-bdan 16410	. . 3 |- BOUNDED ( x e. A /\ ph )
5	4	bdcab 16444	. 2 |- BOUNDED { x | ( x e. A /\ ph ) }
6		df-rab 2519	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	5, 6	bdceqir 16439	1 |- BOUNDED { x e. A | ph }

Syntax hints:   /\ wa 104   e. wcel 2202   {cab 2217   {crab 2514  BOUNDED wbd 16407  BOUNDED wbdc 16435

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213  ax-bd0 16408  ax-bdan 16410  ax-bdsb 16417

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-cleq 2224  df-clel 2227  df-rab 2519  df-bdc 16436

This theorem is referenced by:  bdrabexg  16501