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Theorem ssres2 5583
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 5284 . . 3 (𝐴𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2 sslin 3982 . . 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 5278 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
5 df-res 5278 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
63, 4, 53sstr4g 3787 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3340  cin 3714  wss 3715   × cxp 5264  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-opab 4865  df-xp 5272  df-res 5278
This theorem is referenced by:  imass2  5659  1stcof  7363  2ndcof  7364  tfrlem15  7657  gsum2dlem2  18570  txkgen  21657  funpsstri  31970  resnonrel  38400  mptrcllem  38422  rtrclexi  38430  cnvrcl0  38434  relexpss1d  38499  relexp0a  38510  supcnvlimsup  40475
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