Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdrabexg GIF version

Theorem bdrabexg 15398
Description: Bounded version of rabexg 4172. (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 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem bdrabexg
StepHypRef Expression
1 ssrab2 3264 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 bdrabexg.bdc . . . 4 BOUNDED 𝐴
3 bdrabexg.bd . . . 4 BOUNDED 𝜑
42, 3bdcrab 15344 . . 3 BOUNDED {𝑥𝐴𝜑}
54bdssexg 15396 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
61, 5mpan 424 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2164  {crab 2476  Vcvv 2760  wss 3153  BOUNDED wbd 15304  BOUNDED wbdc 15332
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15305  ax-bdan 15307  ax-bdsb 15314  ax-bdsep 15376
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166  df-bdc 15333
This theorem is referenced by:  bj-inex  15399
  Copyright terms: Public domain W3C validator