Theorem bdrabexg 16199

Description: Bounded version of rabexg 4226. (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 3309	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		bdrabexg.bdc	. . . 4 ⊢ BOUNDED 𝐴
3		bdrabexg.bd	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	bdcrab 16145	. . 3 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
5	4	bdssexg 16197	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
6	1, 5	mpan 424	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2200  {crab 2512  Vcvv 2799   ⊆ wss 3197  BOUNDED wbd 16105  BOUNDED wbdc 16133

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16106  ax-bdan 16108  ax-bdsb 16115  ax-bdsep 16177

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-bdc 16134

This theorem is referenced by:  bj-inex  16200