Theorem bdrabexg 15775

Description: Bounded version of rabexg 4186. (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 3277	. 2 |- { x e. A | ph } C_ A
2		bdrabexg.bdc	. . . 4 |- BOUNDED A
3		bdrabexg.bd	. . . 4 |- BOUNDED ph
4	2, 3	bdcrab 15721	. . 3 |- BOUNDED { x e. A | ph }
5	4	bdssexg 15773	. 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 2175   {crab 2487   _Vcvv 2771   C_ wss 3165  BOUNDED wbd 15681  BOUNDED wbdc 15709

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-bd0 15682  ax-bdan 15684  ax-bdsb 15691  ax-bdsep 15753

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492  df-v 2773  df-in 3171  df-ss 3178  df-bdc 15710

This theorem is referenced by:  bj-inex  15776