Theorem bdrabexg 16944

Description: Bounded version of rabexg 4279. (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 3333	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		bdrabexg.bdc	. . . 4 ⊢ BOUNDED 𝐴
3		bdrabexg.bd	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	bdcrab 16890	. . 3 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
5	4	bdssexg 16942	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
6	1, 5	mpan 428	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  {crab 2532  Vcvv 2821   ⊆ wss 3220  BOUNDED wbd 16850  BOUNDED wbdc 16878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16851  ax-bdan 16853  ax-bdsb 16860  ax-bdsep 16922

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-in 3226  df-ss 3233  df-bdc 16879

This theorem is used by:  bj-inex  16945