Theorem bdcrab 14689

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 14683	. . . 4 |- BOUNDED x e. A
3		bdcrab.2	. . . 4 |- BOUNDED ph
4	2, 3	ax-bdan 14652	. . 3 |- BOUNDED ( x e. A /\ ph )
5	4	bdcab 14686	. 2 |- BOUNDED { x | ( x e. A /\ ph ) }
6		df-rab 2464	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	5, 6	bdceqir 14681	1 |- BOUNDED { x e. A | ph }

Syntax hints:   /\ wa 104   e. wcel 2148   {cab 2163   {crab 2459  BOUNDED wbd 14649  BOUNDED wbdc 14677

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14650  ax-bdan 14652  ax-bdsb 14659

This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-rab 2464  df-bdc 14678

This theorem is referenced by:  bdrabexg  14743