Theorem bdrabexg 13788

Description: Bounded version of rabexg 4125. (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 3227	. 2 |- { x e. A | ph } C_ A
2		bdrabexg.bdc	. . . 4 |- BOUNDED A
3		bdrabexg.bd	. . . 4 |- BOUNDED ph
4	2, 3	bdcrab 13734	. . 3 |- BOUNDED { x e. A | ph }
5	4	bdssexg 13786	. 2 |- ( ( { x e. A | ph } C_ A /\ A e. V ) -> { x e. A | ph } e. _V )
6	1, 5	mpan 421	1 |- ( A e. V -> { x e. A | ph } e. _V )

Syntax hints:   -> wi 4   e. wcel 2136   {crab 2448   _Vcvv 2726   C_ wss 3116  BOUNDED wbd 13694  BOUNDED wbdc 13722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bd0 13695  ax-bdan 13697  ax-bdsb 13704  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-bdc 13723

This theorem is referenced by:  bj-inex  13789