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Theorem resdisj 6013
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 5835 . 2 ((𝐴𝐵) = ∅ → (𝐶 ↾ (𝐴𝐵)) = (𝐶 ↾ ∅))
2 resres 5853 . 2 ((𝐶𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴𝐵))
3 res0 5844 . . 3 (𝐶 ↾ ∅) = ∅
43eqcomi 2833 . 2 ∅ = (𝐶 ↾ ∅)
51, 2, 43eqtr4g 2884 1 ((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cin 3918  c0 4275  cres 5544
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-opab 5115  df-xp 5548  df-rel 5549  df-res 5554
This theorem is referenced by:  fvsnun1  6932
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