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Theorem resdisj 5803
 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 5623 . 2 ((𝐴𝐵) = ∅ → (𝐶 ↾ (𝐴𝐵)) = (𝐶 ↾ ∅))
2 resres 5645 . 2 ((𝐶𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴𝐵))
3 res0 5632 . . 3 (𝐶 ↾ ∅) = ∅
43eqcomi 2833 . 2 ∅ = (𝐶 ↾ ∅)
51, 2, 43eqtr4g 2885 1 ((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∩ cin 3796  ∅c0 4143   ↾ cres 5343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-opab 4935  df-xp 5347  df-rel 5348  df-res 5353 This theorem is referenced by:  fvsnun1  6701  fvsnun1OLD  6703
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