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Theorem resdisj 6168
Description: A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
resdisj ((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)

Proof of Theorem resdisj
StepHypRef Expression
1 reseq2 5976 . 2 ((𝐴𝐵) = ∅ → (𝐶 ↾ (𝐴𝐵)) = (𝐶 ↾ ∅))
2 resres 5994 . 2 ((𝐶𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴𝐵))
3 res0 5985 . . 3 (𝐶 ↾ ∅) = ∅
43eqcomi 2741 . 2 ∅ = (𝐶 ↾ ∅)
51, 2, 43eqtr4g 2797 1 ((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cin 3947  c0 4322  cres 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682  df-rel 5683  df-res 5688
This theorem is referenced by:  fvsnun1  7179
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