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Theorem ssres2 6022
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 5704 . . 3 (𝐴𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2 sslin 4243 . . 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 5697 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
5 df-res 5697 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
63, 4, 53sstr4g 4037 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3480  cin 3950  wss 3951   × cxp 5683  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-opab 5206  df-xp 5691  df-res 5697
This theorem is referenced by:  imass2  6120  1stcof  8044  2ndcof  8045  tfrlem15  8432  gsum2dlem2  19989  txkgen  23660  funpsstri  35766  eldisjss  38739  resnonrel  43605  mptrcllem  43626  rtrclexi  43634  cnvrcl0  43638  relexpss1d  43718  relexp0a  43729  supcnvlimsup  45755
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