Theorem bdrabexg 14518

Description: Bounded version of rabexg 4145. (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 3240	. 2 |- { x e. A | ph } C_ A
2		bdrabexg.bdc	. . . 4 |- BOUNDED A
3		bdrabexg.bd	. . . 4 |- BOUNDED ph
4	2, 3	bdcrab 14464	. . 3 |- BOUNDED { x e. A | ph }
5	4	bdssexg 14516	. 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 2148   {crab 2459   _Vcvv 2737   C_ wss 3129  BOUNDED wbd 14424  BOUNDED wbdc 14452

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14425  ax-bdan 14427  ax-bdsb 14434  ax-bdsep 14496

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-bdc 14453

This theorem is referenced by:  bj-inex  14519