Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xpdisjres Structured version   Visualization version   GIF version

Theorem xpdisjres 29974
Description: Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Assertion
Ref Expression
xpdisjres ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)

Proof of Theorem xpdisjres
StepHypRef Expression
1 df-res 5367 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 xpdisj1 5809 . 2 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ∅)
31, 2syl5eq 2825 1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  Vcvv 3397  cin 3790  c0 4140   × cxp 5353  cres 5357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-opab 4949  df-xp 5361  df-rel 5362  df-res 5367
This theorem is referenced by:  padct  30063
  Copyright terms: Public domain W3C validator