Theorem bdrabexg 9291

Description: Bounded version of rabexg 3891. (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 3019	. 2 ⊢ {x ∈ A ∣ φ} ⊆ A
2		bdrabexg.bdc	. . . 4 ⊢ BOUNDED A
3		bdrabexg.bd	. . . 4 ⊢ BOUNDED φ
4	2, 3	bdcrab 9241	. . 3 ⊢ BOUNDED {x ∈ A ∣ φ}
5	4	bdssexg 9289	. 2 ⊢ (({x ∈ A ∣ φ} ⊆ A ∧ A ∈ 𝑉) → {x ∈ A ∣ φ} ∈ V)
6	1, 5	mpan 400	1 ⊢ (A ∈ 𝑉 → {x ∈ A ∣ φ} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1390  {crab 2304  Vcvv 2551   ⊆ wss 2911  BOUNDED wbd 9201  BOUNDED wbdc 9229

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9202  ax-bdan 9204  ax-bdsb 9211  ax-bdsep 9273

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-bdc 9230

This theorem is referenced by:  bj-inex  9292