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Theorem resdisj 6169
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 5977 . 2 ((𝐴𝐵) = ∅ → (𝐶 ↾ (𝐴𝐵)) = (𝐶 ↾ ∅))
2 resres 5995 . 2 ((𝐶𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴𝐵))
3 res0 5986 . . 3 (𝐶 ↾ ∅) = ∅
43eqcomi 2742 . 2 ∅ = (𝐶 ↾ ∅)
51, 2, 43eqtr4g 2798 1 ((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cin 3948  c0 4323  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684  df-res 5689
This theorem is referenced by:  fvsnun1  7180
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