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Theorem res0 4793
Description: A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
Assertion
Ref Expression
res0 (𝐴 ↾ ∅) = ∅

Proof of Theorem res0
StepHypRef Expression
1 df-res 4521 . 2 (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V))
2 0xp 4589 . . 3 (∅ × V) = ∅
32ineq2i 3244 . 2 (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅)
4 in0 3367 . 2 (𝐴 ∩ ∅) = ∅
51, 3, 43eqtri 2142 1 (𝐴 ↾ ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1316  Vcvv 2660  cin 3040  c0 3333   × cxp 4507  cres 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-xp 4515  df-res 4521
This theorem is referenced by:  ima0  4868  resdisj  4937  smo0  6163  tfr0dm  6187  tfr0  6188  fnfi  6793  setsslid  11920
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