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Theorem xpdisjres 28581
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 4944 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 xpdisj1 5364 . 2 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ∅)
31, 2syl5eq 2560 1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  Vcvv 3077  cin 3443  c0 3777   × cxp 4930  cres 4934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-opab 4542  df-xp 4938  df-rel 4939  df-res 4944
This theorem is referenced by:  padct  28673
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