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Theorem resexg 4865
Description: The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
resexg (𝐴𝑉 → (𝐴𝐵) ∈ V)

Proof of Theorem resexg
StepHypRef Expression
1 resss 4849 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssexg 4073 . 2 (((𝐴𝐵) ⊆ 𝐴𝐴𝑉) → (𝐴𝐵) ∈ V)
31, 2mpan 421 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1481  Vcvv 2689  wss 3074  cres 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3080  df-ss 3087  df-res 4557
This theorem is referenced by:  resex  4866  offres  6039  resixp  6633  climres  11102  setsvalg  12021  setsex  12023  setsslid  12041
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