Theorem bdrabexg 15398

Description: Bounded version of rabexg 4172. (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 3264	. 2 |- { x e. A | ph } C_ A
2		bdrabexg.bdc	. . . 4 |- BOUNDED A
3		bdrabexg.bd	. . . 4 |- BOUNDED ph
4	2, 3	bdcrab 15344	. . 3 |- BOUNDED { x e. A | ph }
5	4	bdssexg 15396	. 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 2164   {crab 2476   _Vcvv 2760   C_ wss 3153  BOUNDED wbd 15304  BOUNDED wbdc 15332

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15305  ax-bdan 15307  ax-bdsb 15314  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166  df-bdc 15333

This theorem is referenced by:  bj-inex  15399