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Theorem ssres2 5874
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 5567 . . 3 (𝐴𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2 sslin 4208 . . 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 5560 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
5 df-res 5560 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
63, 4, 53sstr4g 4009 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3492  cin 3932  wss 3933   × cxp 5546  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-opab 5120  df-xp 5554  df-res 5560
This theorem is referenced by:  imass2  5958  1stcof  7708  2ndcof  7709  tfrlem15  8017  gsum2dlem2  19020  txkgen  22188  funpsstri  32905  eldisjss  35851  resnonrel  39830  mptrcllem  39851  rtrclexi  39859  cnvrcl0  39863  relexpss1d  39928  relexp0a  39939  supcnvlimsup  41897
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