MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssres2 Structured version   Visualization version   GIF version

Theorem ssres2 6025
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 5708 . . 3 (𝐴𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2 sslin 4251 . . 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 5701 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
5 df-res 5701 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
63, 4, 53sstr4g 4041 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3478  cin 3962  wss 3963   × cxp 5687  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-opab 5211  df-xp 5695  df-res 5701
This theorem is referenced by:  imass2  6123  1stcof  8043  2ndcof  8044  tfrlem15  8431  gsum2dlem2  20004  txkgen  23676  funpsstri  35747  eldisjss  38720  resnonrel  43582  mptrcllem  43603  rtrclexi  43611  cnvrcl0  43615  relexpss1d  43695  relexp0a  43706  supcnvlimsup  45696
  Copyright terms: Public domain W3C validator