Theorem bdrabexg 11680

Description: Bounded version of rabexg 3980. (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 3106	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		bdrabexg.bdc	. . . 4 ⊢ BOUNDED 𝐴
3		bdrabexg.bd	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	bdcrab 11626	. . 3 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
5	4	bdssexg 11678	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
6	1, 5	mpan 415	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1438  {crab 2363  Vcvv 2619   ⊆ wss 2999  BOUNDED wbd 11586  BOUNDED wbdc 11614

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11587  ax-bdan 11589  ax-bdsb 11596  ax-bdsep 11658

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-in 3005  df-ss 3012  df-bdc 11615

This theorem is referenced by:  bj-inex  11681