Theorem bdrabexg 10026

Description: Bounded version of rabexg 3900. (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 3025	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		bdrabexg.bdc	. . . 4 ⊢ BOUNDED 𝐴
3		bdrabexg.bd	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	bdcrab 9972	. . 3 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
5	4	bdssexg 10024	. 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 1393  {crab 2310  Vcvv 2557   ⊆ wss 2917  BOUNDED wbd 9932  BOUNDED wbdc 9960

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdsb 9942  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961

This theorem is referenced by:  bj-inex  10027