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Theorem resdisj 4751
 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 resres 4624 . 2 ((𝐶𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴𝐵))
2 reseq2 4607 . . 3 ((𝐴𝐵) = ∅ → (𝐶 ↾ (𝐴𝐵)) = (𝐶 ↾ ∅))
3 res0 4616 . . 3 (𝐶 ↾ ∅) = ∅
42, 3syl6eq 2088 . 2 ((𝐴𝐵) = ∅ → (𝐶 ↾ (𝐴𝐵)) = ∅)
51, 4syl5eq 2084 1 ((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∩ cin 2916  ∅c0 3224   ↾ cres 4347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by:  fvsnun1  5360
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