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Theorem ssres2 5868
Description: Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ssres2 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Proof of Theorem ssres2
StepHypRef Expression
1 xpss1 5561 . . 3 (𝐴𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2 sslin 4196 . . 3 ((𝐴 × V) ⊆ (𝐵 × V) → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V)))
31, 2syl 17 . 2 (𝐴𝐵 → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V)))
4 df-res 5554 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
5 df-res 5554 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
63, 4, 53sstr4g 3998 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3480  cin 3918  wss 3919   × cxp 5540  cres 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-opab 5115  df-xp 5548  df-res 5554
This theorem is referenced by:  imass2  5952  1stcof  7714  2ndcof  7715  tfrlem15  8024  gsum2dlem2  19091  txkgen  22260  funpsstri  33065  eldisjss  36076  resnonrel  40208  mptrcllem  40229  rtrclexi  40237  cnvrcl0  40241  relexpss1d  40322  relexp0a  40333  supcnvlimsup  42308
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