Theorem bdrabexg 7129

Description: Bounded version of rabexg 3874. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdrabexg.bd	⊢ BOUNDED φ
bdrabexg.bdc	⊢ BOUNDED A

Assertion

Ref	Expression
bdrabexg	⊢ (A ∈ 𝑉 → {x ∈ A ∣ φ} ∈ V)

Distinct variable group:   x,A

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

Proof of Theorem bdrabexg

Step	Hyp	Ref	Expression
1		ssrab2 3002	. 2 ⊢ {x ∈ A ∣ φ} ⊆ A
2		bdrabexg.bdc	. . . 4 ⊢ BOUNDED A
3		bdrabexg.bd	. . . 4 ⊢ BOUNDED φ
4	2, 3	bdcrab 7079	. . 3 ⊢ BOUNDED {x ∈ A ∣ φ}
5	4	bdssexg 7127	. 2 ⊢ (({x ∈ A ∣ φ} ⊆ A ∧ A ∈ 𝑉) → {x ∈ A ∣ φ} ∈ V)
6	1, 5	mpan 402	1 ⊢ (A ∈ 𝑉 → {x ∈ A ∣ φ} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1374  {crab 2288  Vcvv 2535   ⊆ wss 2894  BOUNDED wbd 7039  BOUNDED wbdc 7067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bd0 7040  ax-bdan 7042  ax-bdsb 7049  ax-bdsep 7111

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-in 2901  df-ss 2908  df-bdc 7068

This theorem is referenced by:  bj-inex  7130