Theorem bdrabexg 16227

Description: Bounded version of rabexg 4226. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdrabexg.bd	|- BOUNDED ph
bdrabexg.bdc	|- BOUNDED A

Assertion

Ref	Expression
bdrabexg	|- ( A e. V -> { x e. A | ph } e. _V )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem bdrabexg

Step	Hyp	Ref	Expression
1		ssrab2 3309	. 2 |- { x e. A | ph } C_ A
2		bdrabexg.bdc	. . . 4 |- BOUNDED A
3		bdrabexg.bd	. . . 4 |- BOUNDED ph
4	2, 3	bdcrab 16173	. . 3 |- BOUNDED { x e. A | ph }
5	4	bdssexg 16225	. 2 |- ( ( { x e. A | ph } C_ A /\ A e. V ) -> { x e. A | ph } e. _V )
6	1, 5	mpan 424	1 |- ( A e. V -> { x e. A | ph } e. _V )

Syntax hints:   -> wi 4   e. wcel 2200   {crab 2512   _Vcvv 2799   C_ wss 3197  BOUNDED wbd 16133  BOUNDED wbdc 16161

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16134  ax-bdan 16136  ax-bdsb 16143  ax-bdsep 16205

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-bdc 16162

This theorem is referenced by:  bj-inex  16228