Theorem bdrabexg 14218

Description: Bounded version of rabexg 4141. (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 3238	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		bdrabexg.bdc	. . . 4 ⊢ BOUNDED 𝐴
3		bdrabexg.bd	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	bdcrab 14164	. . 3 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
5	4	bdssexg 14216	. 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 2146  {crab 2457  Vcvv 2735   ⊆ wss 3127  BOUNDED wbd 14124  BOUNDED wbdc 14152

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-bd0 14125  ax-bdan 14127  ax-bdsb 14134  ax-bdsep 14196

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-v 2737  df-in 3133  df-ss 3140  df-bdc 14153

This theorem is referenced by:  bj-inex  14219